The reason we use flash is that flash is very bright (and very fast) for easy camera exposure when light is needed. In comparison, the brightest household light bulbs are dim for photography, not nearly sunlight bright. Without flash, even reasonably lighted rooms will suffer from needing long slow shutter speeds, or high ISO, or probably both. Light bulbs are continuous light, which can be fine for still life photography, when a one second shutter speed is no problem (on a tripod), but which is unacceptable for pictures of people, who tend to move. Flash is about instantaneously fast (freezing motion) which also allows us to create the lighting, to add it like we want it to be, to place flash wherever we want the light, and to be made soft as we might desire, etc.
Flash photography is many things. There is on-camera flash and off-camera flash, manual flash and automatic TTL flash, and direct flash and bounce flash. There is fill flash in bright sun, multiple flash units, studio and portrait and table top flash in umbrellas, high speed flash, and more. Lighting is a big and fun subject, but before anyone can get much into "lighting", there are a few more fundamental basics we need to know, about "light". In all of these cases, there are basic differences between flash and existing continuous ambient light. Unfortunately, one downside of automation today is that many photographers have not bothered to learn the actual basics. They can only get whatever their camera does (and cameras are much dumber than we might imagine). But knowing what to expect is the biggest part of knowing how to deal with it. When we know why, then we know how, and then it's easy. Flash is not difficult, it is just different than either sunlight or regular continuous room light, and we need to understand flash too.
In short summary, the major points, the basics you need to understand, the really big deal about using flash, is:
But in drastic contrast, sunlight is quite unique, very special because the Sun is so extremely distant (93 million miles) that its intensity appears not to vary with distance (intensity does vary with distance, but we can't detect it here on Earth, where everything is at the same distance from the Sun). BUT DO NOT assume flash acts the same — it doesn't. Flash is typically in the same room with us, and doubling its distance to subject reduces the flash exposure by 2 EV, which is a LOT.
The speedlight flash can be extremely fast (awesomely so when at lower power levels needing close range), much faster than any possible shutter speed, easily stopping extreme motion (like water drop splashes, hummingbird wings, or even bullets in flight). At Full power level, speedlights are considerably less fast (but still faster than the shutter speed), but even at half power level, speedlights are typically only 1/1000 second duration (and perhaps 1/30000 second at 1/64 power, called speedlights). Studio flash units are typically fast enough too, but camera speedlight flash units can be extremely faster. Speedlights are how high speed flash photography is done (see an example).
However HSS Mode (High Speed Sync) is an extreme exception. HSS flash is a longer continuous duration, dependent on shutter speed, but independent of sync speed (continuous light is always in "sync"). The HSS flash mode is intentionally made to be continuous, meaning present when the shutter opens, and still present when the shutter closes (like sun light is always present). So as seen by the shutter, HSS acts exactly like continuous light (it is continuous for the full shutter duration). And like any continuous light (sunlight, incandescent, etc), HSS is affected by shutter speed, like we always understood for continuous light. HSS has no shutter sync speed limit, but continuous light has no speed at all, all it has going is shutter speed. Speedlight mode can be much faster than any shutter, but is limited by a maximum flash sync speed, where HSS is not. Fast HSS shutter speed is usually used to achieve a wide open aperture with flash in bright sun. But to be continuous, HSS is limited in flash power, so otherwise, normal speedlight flash mode (Not HSS) has all of the advantages (power, range, speed), and is typically much faster than the shutter, and so does not care about shutter speed (shutter merely has to be open to pass it).
But since shutter speed does affect any continuous ambient light, then specifically (except HSS), we can use shutter speed as an important tool to adjust the ratio between regular flash and continuous ambient in our photos. We might choose to totally shut ambient out (overpowering it with flash), or we can include a bit of flash to help soften the harsh ambient shadows.
Also rapid shooting has to wait up to a second or two for Recycle time between shots, to recharge to be Ready for the next shot. Very much rapid shooting also has dangerous overheating possibilities which can damage the flash. Studio lights are normally fan cooled.
This really is pretty easy stuff, but it can be new ideas too. So I suggest that it could obviously help to try just a little before letting your eyes glaze over. :) Photographers simply need to know. Many of us do just "want the picture" without any bother. But we are either interested in the photography or we're not (and there's much hope for those who are interested). It's about like learning to drive — we did have to learn to operate the car, and we did have to learn the rules of the road, and maybe a little about maps, and ice and snow too. So we learned a few easy things, and now we've got it for life, and can use it at will to great advantage, in any situation, with great results. So yes, there are a few extra details to know about flash too, and photographers may actually have to think a little about creating their picture now and then, but none of it is difficult. The rewards are great.
If you don't bother to learn anything about flash, then flash pictures may be among your worst results, and enthusiasm is surely diminished. :) But the opposite view is if you try just a little to understand how to use it, your very best pictures may be due to flash. Flash is how we make the light be great, to improve the scene. Being willing to give it a little thought will certainly always help photography.
"Lighting" is a skill to be developed (which is mostly about learning to actually "see" the results, to look, to notice, to actually realize what you see, to think, to actually be aware of what happened). But the flash is just a light, and we can see exactly what it does (if we just bother to look at the result, carefully enough to actually "see" it). Anything you cannot see should not matter in a picture, however, we do learn to observe much more with experience. Much of the problem is that beginners probably have not yet learned to see much, and you should be aware that the ability to "see" is the main thing to be learned about lighting. There is a page about that too. Flash is just additional illumination, however we can aim and modify the light to be like we want it to be, and we can learn to control the shadows it makes. This concern is not so much speaking of the shadows behind the subject, because generally we try to eliminate those. This is speaking of learning to notice and see the shadows and soft gradient tones ON the subject's face, intentionally created by off-camera lighting (including bounce flash).
Flash can be necessary, and/or it can be a big help. The simplest tips for universally better hot shoe speedlight snapshots (the way we ought to be using our speedlights, but sadly, many don't know yet) are:
Which flash? The camera internal flash is a very small flash, extremely minimal, perhaps sometimes better than nothing, and convenient to carry, but it is very limited in use. A regular size hot shoe speedlight will have at least 4x power, at least 2x more Guide Number, and at least 2 stops more range. Other than power limitations, the internal flash won't zoom for longer lenses, it won't tilt for bounce flash, and it can't be used off-camera. Today, there are Chinese import flashes for less than $100 US, some of which are excellent, for example, Yongnuo flashes are very popular. See a Beginners Guide to Speedlights.
You should try these things, the results will be self-evident. There is a detail or two though, and if your experience level is comfortable working with aperture and shutter speed, and if you want to be able to manage your flash pictures, then the material here is what you need to know. If still a beginner, and not yet comfortable, you really do need to know, and there is a very useful book here. These first basics are the dividing line where we become knowledgeable, no longer clueless. We won't be able to go very far without fundamentals.
These flash basics here are about "light itself", and are obviously important even if you always use automatic TTL metered flash.
Some people avoid any math. Inverse Square Law and Guide Number concepts do have a little arithmetic that a user might follow, easy stuff. It will not do you any harm by simply reading it. We do need to understand the concepts, about what to expect. There are numbers in photography, so there are several little calculators scattered around here to help any deeper questions. Most of the following pages involve less calculation, but still, a concern might be "how bright is my flash?"