Red eye occurs when light (particularly from a flash) reflects off the retina of the subjects' eyes at or near the axis of the lens. When subjects are in low light, their eye pupils are required to dilate (open wider) to allow more light into the eye.
The retina is a wide area located at the back of the subject's eyes and continuous with the optic nerve. It consists of several layers and a web of blood-carrying veins. Therefore, the color reflected from humans is red or occasionally brown (depends on light angle). The color will vary with different animals depending on the shape and the natural filtration qualities of their eyes.
See additional information at "Eliminate red eye."