Flash intensity falls off with distance. Guide Number is a numerical method used to determine exposure of direct flash with manual flash power levels, to automatically deal with the Inverse Square Law, making the math be trivial.
The full details explaining guide number are below this Guide Number Calculator.
One special feature here is this: The marked numbers for a camera's settings are nominal numbers (markings are rounded approximate numbers, not exactly the real numbers the camera actually uses). One example is f/11 ... we say f/11, but which is actually f/11.31. This is a small difference, only about 0.08 stop, but it blurs calculations, which cannot show 0.00 accuracy. Any discrepancy is only in the rounded markings, not any actual problem since the camera always knows to do it accurately. Computing GN manually still works well enough of course, because the camera only accepts third stop values, so 0.08 is not a big deal. We always have to choose a third stop anyway, which could be up to 1/6 stop off (0.167). But as an exercise, this calculator also uses the precise values internally (same as the camera does). That's a small difference, but using the exact values allows more precision in the EV calculation (more consistent). I hope this is interesting rather than a confusion.
There are a few ways to use the tool.
Plan A offers two widely different situations (the two rows), simultaneously, but not related to each other. Choose either/or.
But if you enter the second row computed aperture result into the top row aperture, then they are Plan B again, for the selected power level.
The idea of Plan B is to enter both aperture and distance (your goal), and then match the power level as indicated.
Guide Number (GN) is a prime fundamental, related to Inverse Square Law, and is about how light works, something to know, which will always be important to know. The light fall off means that direct flash exposure can be correct at only one specific distance from the flash. Anything closer is brighter, and anything farther is darker. But how much it changes works on sort of an exaggerated percentage basis (inverse square law), and a greater distance simply has more middle ground range. Bounce flash can seem to extend this range, but direct flash exposure falls off with the square of the distance. Flash will be two stops underexposed at twice the distance, or two stops overexposed at half the distance (inverse square law). So the general rule for flash is to keep all of your subject parts near the same distance plane (same idea as focus depth of field).
Photographing groups: Same distance plane is impossible for multiple rows, and multiple rows can be a deep zone for an even flash illumination, or for focus depth of field too. For long rows, curving the ends forward to equalize the distance helps. For large groups of a few rows deep, raising camera and flash height dramatically (with step ladder) to look down into the group can minimize difference of row distances (and won't hide faces with rows in front). Greater flash distance can extend the range of acceptable flash exposure. We normally think umbrellas ought to be "close as possible" for softness, but when back twenty feet, umbrellas don't add much (will cost power, and softness is not important then anyway). However, increasing the flash distance for greater flash exposure range, and stopping the lens down for greater depth of field, very significantly increases the flash power needed. Common notions for best group lighting for multiple rows is that multiple flashes ought to all be above the camera, pointing outward to cover the group evenly (lights at the camera see same as what the lens sees, without creating terrible shadows). Two flashes aimed different directions are individual units, NOT combining the same as multiple flashes ganged acting as one. But be careful about any central overlap, which is ganged. It would be good to meter the lights and the center, to verify all group areas are equal. For large groups, see Google  I'd suggest the Chuck Gardner link there.
This diversion is really about the distance range depth of the flash exposure. Even if you meter the flash, you can meter the range depth too (you certainly ought to plan and know the exposure difference at front and rear of a big group). Or use the calculator here. Or if computing guide number, simply computing distance at ± 1/3 stop apertures computes the range extents for that tolerance (Note ± 1/3 stop is 2/3 stop from front to rear). Exposure range is not about power or aperture, but is about the flash distance (inverse square law), so as to distance, the range distance is applicable to TTL too.
Guide number makes exposure computation very easy. Guide number is the oldest system to determine flash exposure (used for flash bulbs, before automation), but guide number only applies to direct flash. Guide number is not useful for bounce, because it requires knowing the distance in the total path from flash to subject, and also the reflection coefficient at the ceiling (very roughly, common situation bounce can need 2 or 3 stops more power than direct flash). But guide number still is fundamental today, and understanding guide numbers can increase understanding of flash and inverse square law, whether you actually use guide numbers or not. We should all spend a little time playing with this, to understand the concept. It is a genuine basic of flash photography, which simplifies the Inverse Square Law (which is a really huge factor for flash).
Shutter speed is not a factor of flash exposure (Part 2), but f/stop, ISO, flash power, and flash distance are the factors. Distance does not affect our sunlight, but it is pretty tricky for flash. Direct flash exposure falls off with the Inverse Square Law (with distance), a serious complication for determining exposure. If we don't actually meter the flash, then guide numbers can solve distance computation easily (for direct flash). Guide numbers have been calculated forever, at least since first commercial flash bulb about 1930. Guide number was the only system before light meters and electronic automation.
If you meter your flash, either via TTL flash automation, or by using a hand held flash meter, or if you just use the camera's rear LCD and histogram to tweak in your manual flash exposure, then maybe you can get by for awhile without it, but Guide Number certainly does help basic understanding, essential fundamentals of flash that we should know (how flash falls off with distance).
Guide Number works very well for unmodified direct flash (for example, oncamera direct flash). The one big issue is that guide number is not suitable for bounce or umbrellas, etc. So typically, direct bare flash is much less important for studio lights, because we normally heavily modify their light with umbrellas, softboxes, grids or snoots, whatever. This drastically changes their distribution coverage angles, and every change would create very different guide numbers. Guide numbers are typically more common of camera hot shoe speedlights (direct flash), and speedlights do provide specifications for Guide Number as a guide to the flash power and its distance capability (again, it only applies to bare direct flash). Studio lights, not so much about GN, since these normally use various modifiers (umbrellas, softboxes, etc), so they are likely metered.
Guide Number is a tool to determine exposure of direct flash with manual flash power levels, to automatically deal with the Inverse Square Law, making the math be trivial.
Guide Number = f/stop x Distance (those values which actually give a proper exposure)
f/stop = Guide Number / Distance (aperture for other distances)
Distance = Guide Number / f/stop (distances for other apertures)
For any given "correct flash exposure" situation, guide number is simply numerically equal to the aperture number (like the number 8 in f/8) multiplied by the subject distance (like 10 feet). Then for example, the guide number is f/8 x 10 feet = GN 80 (feet units). Specifically, that aperture and distance combination which gives the correct exposure, defines the guide number.
The Distance is from flash to subject. The flash might be on the camera, but the camera position is Not a factor. It is about the flash.
The useful part is that this guide number is a constant for that flash situation, good also for other distances or other apertures. If we know GN for the situation (flash power level and ISO), we can know correct direct flash exposure for any distance or any aperture. This constant GN is initially determined by some trial situation seen to give correct exposure. Or we can use the manufacturers chart of guide number (trial is what they did).
If for example, in any situation at all, if f/8 is seen to give the correct exposure at 10 feet (from the flash), then this defines that the guide number for this situation is determined to be 80 (feet, from f8x10 feet). Whatever situation gives a correct exposure, that determines the actual guide number, by definition.
The overwhelming advantage of knowing this guide number constant is that if we then move the light to be 5 feet from subject, then GN 80 tells us that GN 80 / 5 feet = f/16 will give us correct exposure there too. Or if we open the aperture to f/4, then the correct distance for this flash power will be GN 80 / f4 = 20 feet. This guide number 80 is a constant (in this same flash power situation), for any distance and any aperture, and its purpose is to make the inverse square law be trivial to compute.
Said again From knowing this guide number constant (GN = aperture x distance) for one flash situation (power and spread angle), we can recompute any other aperture/distance combination for correct exposure, which automatically takes the inverse square law into account, involving only the simplest division. For example, if we know the guide number is 80 (feet), then we immediately know that all of these combinations give the same correct flash exposure:
If we know the correct exposure, f/8 at 10 feet = GN 80

Or, if we know the guide number is 80, GN 80 / 10 feet = f/8

You get the idea  any combination computing (f/number x distance) = GN 80 (in this example) also gives the same correct manual flash exposure. The main use is, if our subject is at 14 feet (from the flash), then we know GN 80 / 14 feet = f/5.7. This is a lot to know by simple division, and it really could not be any easier.
Guide Number definition is (distance x f/stop), therefore doubling GN doubles distance range, or doubles actual f/stop Number, which is two more stops either way.
Where do we get this guide number? Whatever aperture and distance that gives an actually correct exposure can compute guide number. Or more commonly, there is also a guide number specification in the flash manuals (see below). Then we only need to know the distance between flash and subject. This guide number is speaking of manual Direct flash, and this guide number will change if you zoom the flash head differently.
Zoom: Zooming the flash head changes the guide number. Zooming in, to match the lens zoom (a more narrow coverage angle), also concentrates the flash power into a more narrow brighter beam appropriate for the lens zoom, with a higher guide number. There will be a guide number chart in the flash manual, with a different guide number for very zoom value. See the sample guide number chart below.
Flashes that do not zoom (like the DSLR camera's internal flash) will have one guide number value. It is printed perhaps as (the Nikon D3200 specification chart):
"Guide Number: Approx. 12/39, 13/43 with manual flash (m/ft, ISO 100, 20 °C/68 °F)"
For manual flash, this says GN 13 (meters) / GN 43 (feet). This implies at full manual power, but we can turn the flash power down as necessary, which lowers the guide number.
You can work in units of either feet or meters. Since there are 3.28 feet in one meter, the GN in feet is simply 3.28 times the GN in meters. Again, see the guide number chart in the flash manual for flashes that zoom (an example chart is below).
Guide number is all we had in the old flash bulb days (and it still works), and before flash units zoomed, they always had a little calculator on them to do this guide number division, but TTL flash mode has made guide numbers less used today. The top few Nikon flashes have a GN Mode, which is a GN calculator (sets flash power level to the aperture and distance). But we can often do the rough math in our heads (if distance is about 10 feet, then GN / 10 = aperture), which often gives a close starting point for proper flash exposure.
The published guide numbers (specs, charts, etc) are for unmodified direct flash and for the specified flash head zoom level. As the speedlight zooms in (longer mm to follow the lens zoom), the reflector concentrates the flash power into a smaller angle that becomes brighter, to cover the same appropriate view that the zoomed lens sees. There will be a different guide number for every zoom setting, and for every power level. Any other reflector situation  lighting modifier (diffusion dome, reflector, bounce, umbrella, whatever)  is a very different guide number. Any other path than direct flash is a different subject (involving longer path and bounce reflection losses, etc).
Guide number makes Inverse Square Law math be easy. The reason this product (of Distance x f/stop) works as a constant for exposure is due to the coincidence that each stop of f/stop numbers increase by the square root of two (1.414) to give half intensity, and the Inverse Square Law distance decreases by the square root of two to give double intensity, and these square factors of 2 offset and cancel in the math, so that the simple product (aperture x distance) is a CONSTANT for correct exposure for this given direct flash situation (ISO, zoom, power level), for any aperture or any distance. It is enough to know that the big deal is that the Guide Number automatically accounts for the Inverse Square Law, making its math be almost trivial for us. This is a big deal, but it is only applicable to bare direct flash.
Or even easier... Flashes compatible with the camera (communication) often know f/stop, ISO and zoom from the hot shoe. So in direct flash Manual mode, they can use their guide number to show the distance calculation (appropriate for the current power level) on their LCD. This can be a fine starting point (again, direct flash only). Can be very helpful.
The guide number charts are typically always printed showing ISO 100 values, and then we know that GN increases by square root of 2, or 1.414x for every doubled step of ISO. Or we divide GN by 1.414 if converting to half of ISO.
ISO  25  50  100  200  400  800  1600  3200  6400 
GN Factor  x0.5  x0.71  x1  x1.41  x2  x2.83  x4  x5.66  x8 
The guide number changes 1.414x for each ISO stop changed 
The flash power level steps of Full, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 are each half power of the previous step. The best fact to know about manual flash is that each half power step is one f/stop of exposure. One stop is a 2x factor, so said again, turning the flash from 1/4 to 1/8 power (which is half) reduces the exposure by one f/stop. This is extremely convenient to know.
Each half power step reduces GN by the square root of 2 (divide GN by 1.414). Two half power steps (1/4 power) is two stops of exposure, or 1/2 the GN value. Or use the calculator, or see the GN chart further below.
Increasing the flash zoom mm number concentrates the power into a smaller beam. Doubling the zoom mm theoretically covers a 1/4 smaller area with 4x brighter intensity, two stops. Which the calculator could calculate, but the actual reflectors vary to much to try. There is substantial area overlap (so frame edges are fully exposed), and usually double zoom mm might multiply guide number about 1.4x, or one stop (if that). Which is only a very rough approximation  because again, of course it depends on the individual reflector design. But guide number maybe about doubles from 24mm to 105mm (4x), which is near a two stop increase (half expected), but there are large variations. I'd love to be able to add zoom to the GN calculator, it is obviously an important factor, but this would have to embed GN charts from many different specific flashes, and there are too many flashes. You can do that easier, so instead see the Guide Number chart in your flash manual.
The GN of multiple equal flashes ganged in combination acting as one, is (GN of one) times square root of (number of flashes). Each double of the number of equal flashes (1,2,4,8 flashes) results in one stop in brightness, which increases GN by the square root of 2 (1.414).
So two equal flashes is 2x which is one stop more, but ganging two unequal flashes, say of GN 56 and GN 80 (one stop difference), should add as Square root of (56² + 80²) = GN 98 (sum about 2/3 stop more than brightest alone)
The GN spec is in the flash manuals, and if it has zoom, it will have many guide numbers... and a GN chart in the manual (for zooms and levels). The charts usually show GN as both meter and feet values (specified as meters/feet). The value for feet is meters/3.28 (3.28 ft = 1 m). Use either feet or meters, so long as you are consistent with units.
The charts are always for ISO 100, but we know the ISO 400 factor is x2. We seek a ISO 400 GN in this example, so to go the other way, we divide the ISO 400 value by this 2x to get the ISO 100 value (to be able to compare in the ISO 100 chart). All of this was just shown above.
Suppose we plan to use direct flash at f/8 at 12 feet at ISO 400. So we obviously know this needs flash power of (f8 x 12 feet) = GN 96 (feet) at ISO 400.
Converting this example (f/8 at 12 feet, GN 96 at ISO 400) to ISO 100 is GN 96/2 = GN 48 (feet, ISO 100). Now we can search the guide number chart in the speedlight manual (ISO 100), and maybe we find this value at 24mm zoom and 1/4 flash power to be GN 49 feet. More than close enough to 48. The point of course is that same chart value also applies to GN 96 at ISO 400. This SB800 GN chart below (for an example) says that 24mm flash head zoom and 1/4 flash power is GN 49, almost exactly the GN 48 that we seek. Measure the distance, set the settings (flash 24mm zoom and manual 1/4 power, camera ISO 400 and f/8), press the shutter button, and Presto, you've got it. You're very close on first try for a 12 foot flash distance (direct flash).
Or, the chart includes several other combinations of power level and zoom which are near GN 49. It need not be exact, another GN value of about 12% difference is roughly within 1/3 stop (and 41% is one stop). We don't necessarily have to match flash zoom to lens zoom values, we can use any wider flash zoom  which may waste a little light if the flash is wider than the lens, which is no problem if not at maximum power (some flashes do not zoom anyway). FX flash on DX cameras is already 1.5x wider anyway. To adjust results of this method, you can simply adjust the power level by 1/3 stop, or adjust the aperture by 1/3 stop. Or zooming the flash head makes steps sometimes roughly about 1/3 stop (no bets, that is a very rough approximation).
Say you really wanted to use 35mm zoom... or that you could use 35mm zoom. You can of course simply select any nearby zoom and power cell near 48 there (say 35mm, GN 44 at 1/8 power, that's close). Multiply it back to ISO 400 (44 * 2 = GN 88), and then compute a similar distance and aperture for that GN value (GN 88/12 feet = f/7.3) at ISO 400, 35mm and 1/8 power. That is comparable to 12 feet at f/8. All that is left is to measure off the 12 feet. There is some situation that will work, and it will work (if using your own chart of course).
Or the GN calculator above works: Choose any zoom in your GN chart, and enter its full power GN (GN 125 here for this 35mm case) at ISO 100. Enter the new ISO 400 you want to use, and for this example, enter f/8 and 12 feet. Click Compute, and it will advise setting 2 2/3 stop power (1/3 higher than 1/8 power). Do that and click Compute again, and it verifies it is within 0.09 stop. And results should be pretty close.
Or if you have it, use the GN mode (next below), set ISO 400 and f/8 on the camera, and 12 feet in the GN menu. It will set the power for 12 feet, and it will work too (feet or meters, whichever you have the flash set to use).
This example chart is from the Nikon SB800 flash manual.
Measured GN sometimes may seem to vary a little from the manufacturer's chart value. I have wondered at times, but prefer to believe the chart numbers are carefully prepared. One factor is that our own procedures can vary. A near surface can provide reflected fill (affecting exposure), so GN can increase in a small room where reflections from the near walls/ceiling combine, whereas GN is lower in wide open spaces with no reinforcement. How much the ambient light is also contributing can be a factor. New flash capacitors do of course have a capacity tolerance (typically 10%, +20%), so individual flash units might vary slightly (but small in terms of stops of power). Another possibility is that the capacitors in an old flash may have deteriorated somewhat, not still full capacity. My 25 year old Nikon SB24 still performs to GN spec however.
The GN charts generally seem to specify GN for meters always in whole numbers, and then GN for feet is simply m/3.28. Just my assumption, but if every zoom value on every brand and model has a rounded GN value in meters (no decimal points), it suggests the published GN precision is ± 0.5 (meters) or ± 1.6 (feet), which is possibly a noticeable variation if subject is close. But if at usual distances, GN can be surprisingly accurate, an excellent first try. Just try it, especially if your flash has the GN mode (next below). I don't have much trouble with guide numbers, but there certainly are variations in different situations..
But I really think (in my own case) the most likely reason for GN differences is that our own perception of "correct exposure" is not always precise, so check multiple tries in different situations. And actually do take the picture to check it. My own experience is that the Nikon guide number seems accurate. If I measure the distance and take the picture, I get fairly good exposures, at least a good starting point. One beauty of it is that GN is not affected by subject reflectance in reflected meters. But if I just meter the direct flash, it may not always match GN expectations precisely? That's a contradiction, and is probably my procedural error? Metering flash is a good thing to do, and if you can meter your lights, that's great. But if you have no other means to adjust manual direct flash, you ought to try guide numbers.
One special case: If you attempt to verify your speedlight's guide number at maximum power level, don't use your maximum sync speed (at full power). Unless you are metering it, then at full power, back off to maybe 1/160 second. The speedlight becomes slow at maximum power, so if at the fastest shutter speed, the shutter can close on it, and the sensor can't see it all. If metering it with a handheld meter, and flash is on the hot shoe, the fastest shutter speed can quench the flash off too, so an external meter can't see it all either. This is only a slight effect and only occurs at maximum flash power and at fastest sync shutter speed. Flashes vary, but I do see my SB800 slightly reduce maximum flash exposure at 1/320 second, but it seems negligible at 1/250 second. Speedlight lower power levels are greatly faster of course. Most studio monolights are the opposite, slowest at lower power levels.
SB700 manual page C11
SB800 manual page 44 (shown at right)
SB900 manual page D11
SB910 manual page C12
Nikon calls this GN mode "Distance Priority Manual Flash". When on the hot shoe, CLS flash already knows ISO, aperture, zoom, and the guide number chart, so these flash models have a GN Mode option where all you do is set the distance into the flash menu (ten feet shown here). The flash computes and sets the flash power level automatically, to be correct for the distance and camera settings (bare direct flash). Automatic computation in that sense, but it is a Manual flash mode  we enter the distance manually (The Dlens focus distance is not used by this mode, because it is not accurate or complete enough for this).
The details are that we know f/8 at 10 feet is 8x10 = GN 80 at this ISO 200. That's 80/1.4 = GN 57 at ISO 100. This is a SB800, so looking at its ISO 100 GN chart above, we know 50mm at GN 57 must be set at a bit more than 1/8 power in this case. GN Mode simply knows how to do all that, and does it for you when you enter ten feet.
Guide Numbers are about direct flash, but this GN mode can surprise you if you will try it. We do have to know the distance, but one really wonderful GN advantage is that unlike TTL metering, GN Mode is independent of the subject colors which do affect TTL metering according to how well they reflect light. GN mode is independent of the subjects reflectance (like incident metering is also independent of seeing the subject). It just sets the right light level for any subject (at that one distance), and black things will come out black, and white things will come out white. So this mode would be fabulous, except that it is direct flash only, and we have to know distance. GN mode is for camera mode A or M (aperture is set and does not keep changing), and for direct flash only with the flash head straight ahead (the Nikon GN mode simply disappears from the menu if the flash head is tilted or rotated).
Guide Numbers do not work so well for bounce, and bounce is often the very good stuff. TTL is wonderful for bounce, but bounce will need two or three stops more flash power then the direct distance GN would indicate. In the real world, we usually just guess at the direct distance, so the initial result might be off a little, but like TTL, it will be a close starting point. Camera Flash Compensation is not operative in GN mode (camera metering is not involved), but the Nikon flashes can use the compensation in the flash body  press the Center SEL button, see the manual. Or you can reenter a different adjusted distance (changing the other values like aperture or ISO just recalculates).
Guide Numbers are used for direct bare flash, but it becomes tough and unknown for bounce and umbrellas, etc. Path distance has to be measured from the light source (the flash tube), via the reflection surface (NOT just from the fabric panel). In the old days (before TTL electronics), we used to approximate for ceiling bounce with the rule of thumb "open two stops for bounce" (from the direct values), which was sometimes adequately ballpark for negative film (much more latitude than digital), but of course, very crude and vague, because every situation was different (ceiling height and texture and reflectance, and flash head angle, etc). This will likely be more like three stops for a vertical flash angle.
Whereas, TTL excels for bounce, it simply meters the actual light arriving via that path, whatever it is. However, regardless if bounce or direct, TTL accuracy is always affected by the reflectivity of the various subject's colors (clothing, walls, etc)  which does not affect guide number, which does not even take the subject into account (only the distance and GN are considered). For direct flash, the beauty is, if we know GN and distance, we KNOW the exposure, independent of the subject colors. There is a good case for that, but frankly, measuring distance and doing division is more awkward than automatic TTL metering. We likely have to adjust it slightly either way (due to TTL reflectance, or GN distance).
So to repeat: Problems are, the guide number method needs to know a fairly precise distance from flash to subject, and this leaves out bounce flash. Or, we can always guess roughly at the distance, and get a rough trial answer, and then tweak that result better by trail and error. It is a good starting point, but we also need to know the guide number fairly precisely, which implies direct flash only. The guide number chart in the flash manual can differ a bit from our results, which could be due to added reflections from walls in a tiny room, could be flash capacitor aging, could be marketing exaggeration of specs. Or, often it is rather accurate. But it will be a constant after you know it.
But if we do know one precise exposure result, we can change distance and still know the right exposure. We have to do some division, but sometimes we can approximate this in our heads, or many flashes have (or used to have) guide number calculators, where we enter distance, and it tells us f/stop, or vice versa.
While guide number is a fundamental basic we ought to know (it handles the Inverse Square Law), and which is still dead on today, frankly, the method may seem old fashioned now. It is what we used back in the 1940s to 1960s  it was all there was for the flash bulb era. But by about 1970, we had electronic flashes with the photo sensors for the Auto modes that selfmetered the reflection back from subject. We certainly liked that, and it worked for bounce too. The 1980s introduced TTL, metered and controlled by the camera computer. We liked that too, it was great to actually meter the flash. Reflective metering certainly can have issues, it often needs some correction (called Flash Compensation, which we add manually by trial and error and experience). But TTL is metered, and is generally always a pretty close starting point.
Many users use TTL flash today, but also many prefer manual flash mode, for the control it offers. Both modes must be watched and adjusted, frankly, both are just the first starting point for determining actual proper exposure. TTL may start closer, but frankly, there is much less difference than we may imagine, in that we adjust both for a final result. Manual flash users just quickly "know" (remember) that this familiar situation will need about 1/4 power, same as last time. Honest, neither method is difficult except first day.
But either way, guide number is really about the least we can know about flash. For example, you're at home wondering about the graduation picture tomorrow. You think you can sit with 50 feet of the stage. You have a suitable lens, say 105mm for DX, but you're wondering about the flash. The zoom on your SB600 maxes out at 85 mm, and the GN there is 131 (ISO 100), and x2 for ISO 400 is GN 262. At 60 feet (safety factor), GN 262 / 60 feet = f/4.4. Piece of cake, ISO 800 should not be necessary. Even if planning to use TTL, this is good to know before you get there. Take a couple shots of the empty stage before things start, to get setup right.
See a page about comparing power ratings of flash units using guide numbers (NOT the same page as Next page below).