Red-eye - Causes, Avoidance, Fixes, And Unusual Occurrences
When a bright light such as a camera flash bulb is fired at a human eye it has sufficient intensity to illuminate the retina at the back of the eyeball and return a red light. The light is red as the mass of the retina is composed of blood vessels and tissue. The light returns via the pupil which is normally perceived as a small black circle at the center of the eye. It is normally black due to the presence of light absorbing pigments. The size of the red-eye effect is determined by the size of the pupil at the moment that the flash is fired. The size of the pupil is determined by the intensity of ambient light present prior to firing the flash. An abundance of light will cause the iris to contract the pupil and thus reduce the amount of light that passes through to the retina. Unfortunately the iris is not quick enough to react to the high intensity burst of a camera flash and the admittance to the retina of this over abundance of light has the effect of illuminating the inside of the eye. To counter the delayed reaction of the iris, cameras provide a feature called 'red eye reduction'. The feature works by firing two flashes, the first one is intended to contract the pupils while the purpose of the subsequent one is to provide sufficient light for a well exposed photograph. In many cases this is sufficient to eliminate the appearance of red-eye.

Pupil And Iris
Avoidance
The only time you will be troubled by the red-eye effect is when you are using the intense light from your flash. You can avoid red-eye entirely by refraining from using your flash. The consequences of shooting without flash though may be even more troublesome as you run the risk of underexposure, motion blur, camera shake, and high levels of noise; all of which are much more difficult to rectify by image processing than would be the relatively minor task of digitally removing the red-eye. You should never have an occurrence of red-eye when shooting in a studio as you should have either ample lighting available to render your flash superfluous or the ambient light intensity should be high enough to contract the subject's pupils sufficiently to reduce the possibility red-eye becoming apparent. If you are using a flash in the studio you are likely to have the ability to use a flash unit that is physically separate from your camera thus by placing the flash to the left or right, the reflected light from the retina will bypass the camera's lens.
Fortunately, it is a very simple matter to repair red-eye. As the problem is so common, all image editing applications support a red-eye removal tool and it is one of the easiest features to use. Usually it is just a matter of selecting the tool and clicking at the center of the pupil; the software will do the rest.
If you don't have any image editing application capable of removing red-eye then you could download a free one such as irfanview, though the results are less than satisfactory in many cases.
Red Eye Close-Up
Fixed Red Eye Close-Up
From the degree of pupil dilation you can discern that this shot was taken in a reasonably dark environment thus the affected area is conspicuously large. In cases where the flash is so intense as to transgress the red channel and give the appearance of yellow or white pupils, the software will fail to deal with the problem and you will have to manually 'paint' a new pupil. You will also have to add some specula highlights to give the pupil the appearance of being reflective.
Yellow Eye
Fixed Yellow Eye
A similar problem to the extremely high intensity of light that produces the white-eye above, is that of the dogs, cats and other species that posses a layer behind their retinas called the tapetum lucidum. This layer reflects light back to the retina and increases their ability to see in low light conditions. This layer will reflect a large amount of the flash light back to the camera lens and produce a white, green or yellow pupil. This effect will have to be removed manually by painting a new pupil.
See Foggy Dog for the full image.
Yellow Dog Eye
Dog Eye Fixed
Five percent of children have a perceptible squint. Often this is satisfactorily compensated for by wearing their prescribed glasses. If they have to remove their glasses for the purpose of an official photograph such as a passport, then the squint will become more apparent. The red-eye will be also more obvious in the affected eye as the wall of the inner eye is closer to the pupil in relation to the axis of the camera lens.
Squint
In this case there is some damage to the iris of the subject's left eye. The affected iris is unable to contract the pupil to compensate for the level of light reaching the retina. The subject's right eye shows us that the level of light is sufficiently high to contract the pupil enough to avoid the occurrence of red-eye when the flash is fired. The red-eye is easily removed and the unevenly sized pupils are consequently less apparent.
Iris Damaged Eye
Two-Shot Red Eye

Discrepancies Between Subjects

In the example on the left we see a shot of two women one of whom exhibits red-eye while the other one doesn't. The woman on the left has large dilated pupils which are naturally prone to this problem. The woman on the right however, has small enough pupils to have avoided the effect.
The reasons behind this apparent discrepancy in pupil size is hidden by the power of the flash itself which has hidden any shadows that may have been there prior to it's firing. In this case there would have been a shadow over the eyes of the woman on the left possibly caused by the photographer himself. This lower light intensity has caused this subject's iris to contract the pupil in order to provide sufficient light for the retina's needs.
People often worry that they have a medical problem when they see that they exhibit red-eye in a photograph where other don't. This is because the evidence that would indicate why this is so has been hidden by the pervasive light of the flash.

Red Eye Pen
Removing red-eye from an actual photographic print can be achieved by the application of a special red-eye pen.

The effect is not as satisfactory as removing the red-eye prior to printing as you can easily perceive the difference between the photographic emulsion and the area on which you've applied the pen, but if you cannot reprint for whatever reason, then one of these pens may be your only option. The pens are available at many photographic stores and they are not very expensive.