Fixing Red Eye
Red eye is the effect caused when a light is reflected off of the blood vessels in the back of the retina wall. This effect is caused when the iris is opened up and gets a quick blast of light it was not ready for. Now that we know the cause we can begin working on the solution.
The easiest way to alleviate this problem is to add more lights to the scene. If you turn a couple of extra lights on in the background it will make the iris contract and lower the amount of light entering the retina. When the iris is contracted the pupil is shrunk and therefore less light can enter, making less light reflect back out.
Using a flash in the hot shoe will raise the flash higher above the lens changing the angle of light and lowering the reflection. There are also extension cords that connect the hot shoe to the external flash unit allowing you to move the flash off to the side and above the camera. Changing the angle not only reduces the amount of reflection but also makes for more pleasing lighting conditions.
Many cameras have a in camera red eye reduction option. This generally is done by sending a quick burst of light before the exposure is taken. What this does is makes the iris contract before the flash is shot for the photograph. Using this option allows you to use the on camera flash.
Using no flash at all will remove the problem, but make getting the photograph will be more difficult. Increasing the ISO of the film or digital sensor will lower your chance of needing a flash, but increases your chance of digital noise. If you have a tripod handy you can set this up and increase your exposure time. Doing this will make it necessary to have your subject sit very still for a longer period of time.
The final option if none of the other can be used is to use photo editing software. Most programs have a red eye repair tool. Generally you just size the tool over the pupil and with black as the selected color you click. These programs most often to an excellent job and make it look like the problem was never there. Try to get the photograph properly the first time but know in worst case scenario this option is here.