Red-eye is caused by light from a flash reflecting off a subject's retinas, taking on the colour of surrounding blood vessels. In some animal eyes, the problem may show up as a green, yellow, or white glare because of a coloured membrane that's located behind the retina.
Man or beast, the problem usually occurs only when the flash is positioned close to the camera lens, as it is on point-and-shoot cameras, and in low lighting, when the subject's pupils are dilated (enlarged) to absorb more light.

If you have no other choice but to use a flash, you can lessen red-eye by taking advantage of the red-eye reduction flash setting found on most cameras.
Another trick is to tell the subject to look slightly up or down so that reflected light from the retinas doesn't bounce back directly into the lens. Of course, moving farther from your subject also reduces red-eye because less flash light reaches the eyes.