Flash
Flashes (or Strobes in pro lingo) are used when you need more light. That doesn't mean quality light, just quantity.

Your flash will have many different settings depending on the camera, but it should have at least the on and off settings:

The "A" is for automatic here.

In addition to these settings, you may have a flash setting that allows you to change the amount of power your flash produces:

This is called flash compensation, and just like your exposure compensation, it allows you to make the power of the flash stronger or weaker manually.

Most cameras have some flash setting that does a pre-flash to dilate the subjects eyes before firing, thus reducing red-eye

This is called red-eye reduction. It doesn't work the same with animals due to animal eye shapes and compositions. Also, don't confuse this with your focus assist light that goes off to help the camera focus in low-light situations. You might want to let everyone know that the camera will flash twice too so they don't think it's gone off when it only preflashed.

More
Depending on the camera, you may have a slow flash (slower shutter speed with normal flash) or even a flash that fires on close of the shutter instead of at opening of the shutter (1st curtain or 2nd curtain).

The main thing to know about flash is that the shutter mechanism is a finite speed, so it can only sync with the flash at a certain speed or slower. If the shutter is too fast, then the shutter would already be closing before the light of the flash had time to bounce off the subject and travel back to the camera; in the film days this would result in half pictures, now the flash just won't let you shoot the picture or it will be super dark. In fact, when you pop-up or turn on the flash, your camera automatically sets the shutter speed to a usable sync speed.

Fill Flash
The problem with flash is that it is so specific in its look and direction. Flash also is very conspicuous meaning that it usually annoys your subject. Flash is best used as the additional light source instead of the main or only light source; for example use it in the daylight to "fill flash" a subject thereby reducing the difference in the light falling on the subject and the light behind the subject.

When using flash for portraits, tell your subjects to close their eyes while you focus and then tell them to open them right before you fire. This helps with blinking, looking the wrong way, and "stoner" half eyes.

Shutter Drag
To achieve a blurry background but a sharp subject, follow the movement of the subject with the camera while firing, if the relative movement of the camera matches the movement of the subject, it will look sharp while the background will have motion blur; this is called Shutter Drag.