Why are your eyes red in pictures?

You are actually seeing images of the retina, the layer of cells at the back of the eye. There are many blood vessels in that part of the eye, as well as a very high concentration of light receptor cells.

You will note that in virtually every photo in which you see 'red eye', the photographer used a flash. The flash was probably on the camera, making it possible for light from the flash to make a reasonably direct trip from flash to retina and back to the camera. The need for the flash suggests that the ambient light level is low enough for the subjects' irises to be open. The flash and exposure happen so quickly that the iris doesn't have time to adjust by closing a little. The result is the slightly creepy effect that we call red eye.

This is why most cameras today have a pre-flash option for photos of people and/or animals. A few short bursts of light cause a person's irises to close a little, and red eye is almost always avoided. If this option is not available with your camera, then placing the flash at an angle, away from the camera, may also help reduce red eye. The problem can also be fixed easily with most photo editing software packages.