Red-Eye Reduction Techniques
First the bad news. The cause of red-eye is a basic fact of human biology--the pupils expand and contract in response to light exposure. In bright light, the pupils are small; in low light they can get really big. When your flash goes off, it travels through the dilated pupil and reflects light off of the blood vessels behind the retina inside the eye. It is reflected back at the camera in the form of a distracting red spot. This is, unfortunately, always the case.
Now the good news. You can use techniques to reduce the flash's effect. Some work better than others. Try them to find which works best for you.
Use red-eye reduction. The most obvious solution is to activate the "red-eye reduction" feature on your camera. When this is used, the camera fires a series of pre-flashes (quick bursts of light) or a steady stream of light that makes the pupil close down before the main flash is fired. When the flash finally does fire as the picture is taken, the pupil should be smaller and therefore reflect less of the inner eye back at the camera. This is often a big help, but keep in mind this is merely red-eye reduction, not removal.
Pre-flashes delay the actual firing of the shutter by as much as one second or more, so if you are trying to capture a baby's first steps or other brief moment, you may be better off with red-eye than missing the moment altogether.
To reduce red-eye, have people look at the opposite side of the camera, away from the flash.
Turn the lights on. Red-eye is worst in dark rooms, so turn on a light or move the subject to a brighter area. The pupils become smaller and the red-eye reaction is reduced.
Look away from the lens. Some cameras are more susceptible to the red-eye effect than others. A flash that is located close to the lens, versus one that pops up or away from the camera, produces more of a reaction because the light directly hits the pupil. Have the subject look away from the lens, either above the camera or to the side opposite the flash, to reduce the reflection.
Get a removable flash. The best solution for red-eye is to move the flash away from the lens altogether. This is only possible on cameras that have a hot shoe or other connection that allows for the addition of an external flash (one that is not built into the camera). This can be a pricey solution, but worth investigating if you are shooting a considerable amount of photos in red-eye prone situations.