Why do people have red eyes in flash photographs?

We've all see photographs where the people in the picture
have spooky red eyes. These are photos taken at night with
a flash. Where do the red eyes come from?

The red color comes from light that reflects off of the
retinas in our eyes. In many animals, including dogs, cats
and deer, the retina has a special reflective layer called
the tapetum lucidum that acts almost like a mirror at the
backs of their eyes. If you shine a flashlight or headlights
into their eyes at night, their eyes shine back with bright,
white light. Here is what Encyclopedia Britannica has to say
about the tapetum lucidum:

Among many nocturnal vertebrates the white compound
guanine is found in the epithelium or retina of the eye. This
provides a mirrorlike surface, the tapetum lucidum, which
reflects light outward and thereby allows a second chance for
its absorption by visual pigments at very low light
intensities. Tapeta lucida produce the familiar eyeshine of
nocturnal animals.

­ Humans don't have this tapetum lucidum layer in their
retinas. If you shine a flashlight in a person's eyes at
night, you don't see any sort of reflection. The flash on
a camera is bright enough, however, to cause a reflection off
of the retina -- what you see is the red color from the blood
vessels nourishing the eye.

Many cameras have a "red eye reduction" feature. In these
cameras, the flash goes off twice -- once right before the
picture is taken, and then again to actually take the picture.
The first flash causes people's pupils to contract, reducing
"red eye" significantly. Another trick is to turn on all the
lights in the room, which also contracts the pupil.

Another way to reduce or eliminate "red eye" in pictures is
to move the flash away from the lens. On most small cameras,
the flash is only an inch or two away from the lens, so the
reflection comes right back into the lens and shows up on the
film. If you can detach the flash and hold it several feet
away from the lens, that helps a lot. You can also try
bouncing the flash off the ceiling if that is an option.