Many digital cameras have a variety of flash modes. They usually include flash on where the camera fires the flash whenever the exposure system decides it is necessary. The light is quite harsh, making the subject stand out against a dark background.
Forced on always fires the flash when you press the shutter, regardless of the amount of available light present while flash off turns the flash off, so you can capture an image with available light without the flash firing.
Anti red-eye is used to reduce red-eye when taking a picture of a person looking straight at the camera. Red-eye is caused by the flash being too close to the lens – as of course all built in flashes are - reflecting the light off the retina at the back of the eye. A pre-flash makes the eye's pupil contract and lessens the red-eye effect. It is not very effective though. A way of avoiding red-eye completely is to make sure the subject is not looking straight into the camera, which will make for a more natural looking portrait anyway.
Slow synch or fill-in mode is the most versatile. The camera will choose a shutter speed for the ambient light plus a small dose of flash to lighten the foreground. This will make for a balanced picture as the subject will be lit by the flash while the background will be recorded as well. Fill-in is also very effective at lightening up dark shadows in harsh sunlight.