Just to add on in case anyone is interested in the biological aspect of it...
The red-eye is actually a result of the light from the flash bouncing off the inside off your eyeball. Thus, in red-eye reduction mode, a pre-flash is fired to give your pupils time to contract (in photographic terms... stop down :bsmilie
, so that when the main flash is fired, not too much of it goes inside your eye, thus you see much less of the red-eye effect.