What causes a red eye in a photo with a flash?
Your camera is recording the reflection from the blood vessels of the retinas at the back of your subject's eyes. The phenomenon is made worse in dark conditions because your subject's irises are open wide, exposing more of the blood-rich retinas.
The phenomenon is not a function of flash as such, but rather of the angle of reflection. There is a physical law which states that the angle of reflection is equal to the angle of incidence. In other words, light bounces off a reflective surface at an angle the same as the angle the light came from. On most compact cameras, the flash is sitting right next to the lens, so the angle of incidence (where the light comes from) is essentially the same as the angle of reflection. The light bounces straight back into the lens. Since the back of the eyeball is richly supplied with blood vessels, the color of the reflection is red and you get those eerie vampire eyes.
What can you do? If you have a flash which can detach from your camera on a cord, you can remove the flash and hold it a foot or more away from the lens. The reflection will go off on the opposite angle and there will be no red eye. But most compact cameras have fixed flashtubes sitting right next to the lens. Manufacturers have devised all sorts of redeye reduction schemes. One of the most common is the rapid repeated low-power firing of the flash before the main flash. This causes the subject's irises to react by closing down, reducing the reflection from the retinas. If you don't have redeye reduction, try raising the light level in the room. If that spoils the mood, try having your subject look into a nearby lamp just before taking the picture. Your subject can also look slightly away from the lens.
If none of that works, or spoils the spontaneity, there are redeye removal pens to retouch your prints. On your computer there may be redeye removal tools. If you have any version of PhotoShop, redeye removal is a cinch.
Most cameras have that funny pre-flash flicker before the flash; that is so that eyes have a chance to adjust, and avoid the red-eye effect. Serious amateurs and professionals avoid red eye by not having the flash so close to the axis of the lens. This is the primary cause, which unfortunately, many people can't do much about with their simpler cameras.