Flash adds light to an image and many cameras feature several flash modes for you to pick from which include:
This is where the camera will use the flash when it detects low light levels. You don't have to tell the camera you want to use the flash, it will fire on its own.
This is when you tell the camera to fire the flash, regardless of the available light. It can be handy when shooting outdoors when there may be darker shadowed areas but generally the lighting is good.
This turns the flash off completely. It's used when available light levels are sufficient or when shooting in areas where flash photography isn't allowed.
Red-eye is usually caused when shooting with a camera that has built-in flash and it makes the eyes look like they have a demonic red glow. By using red-eye reduction, the flash fires several times just before the actual photo is taken. By doing this, the pupils contract and it is less likely you'll have red-eye appearing in the photograph. Do warn your subject before you fire the flash though as it can make them look like a startled deer in headlights if not.
Fill-in Flash is weaker than regular flash, but is strong enough to add light to darker areas of the image such as shadows. It's useful for situations when the subject needs illuminating but the background doesn't, such as the images here:
Left: Without fill-in flash the subject is too dark. Right: With fill-in flash we can now see the subject.
Slow sync flash tells your camera to use a slow shutter speed and to fire the flash. This means your subject will be sharp but you'll still have some light in the background of your shot. In other words, it records the background and then illuminates the subject with flash to balance the exposure between the two areas. It's a good mode to use when shooting portraits at night although a tripod or steady support may be necessary to keep the background sharp.
Front Curtain Sync
This mode tells your camera to fire the flash at the start of the exposure then the shutter will remain open, continuing to record the ambient light.
Rear Curtain Sync
Rear curtain sync tells your camera to fire the flash just before the photograph is finished exposing. This creates a long trail but a nice, crisp shot of your subject. Rear curtain sync is great for adding impact to an action shot as when your subject moves through the scene they'll blur but the pop of flash at the end will leave them crisp and sharp.