You may have noticed that red eye only appears in pictures taken using a flash. The reason for this is simply that the flash causes red eye.
Humans have a curved plate covering the back of their eyeballs called the retina. This is the part of the eye that actually detects the presence of light. When the flash on a camera fires, the bright, white light emitted by the bulb travels into the eye, and bounces off the retina back into the camera lens.
If that was the entire story though, we would call it 'white eye'. While the light is inside our eye, it travels through many tiny blood vessels that are present to keep the eye alive. This is how the light gets its red color.
Of course, it takes a very powerful light to travel through these blood vessels, and return with enough intensity to be visible. If you shine a torch in someone's eye, it does not appear red for example. It is only the powerful flash bulb that makes this effect so visible.
Red Eye Reduction
Some cameras which do have the flash close to the lens also contain a red eye reduction feature. This feature causes the camera to emit two large flashes, one as the picture is taken and one just before. Some cameras even produce a series of minor flashes before the main flash.
These 'pre-flashes' cause the subject's pupils to contract, which reduces the light which can travel through the pupil, to be reflected back into the camera. Even turning up room lights can reduce red eye, as in a brighter room the person's pupils will contract.