Red eyes in an image is the undesirable consequence of reflected light from an artificial light source (usually an electronic flash) that is located too close to the camera. Eye-shine is similar to red-eye but is useful because it helps to locate animals that are so well camouflaged they would otherwise be invisible.
The causes and solutions to avoid red-eye are shown in the diagrams below. Creating a wider angle between the light paths from the flash to the eyes and from the eyes to the camera lens will often prevent red-eye. I prefer getting close to the animal but for obvious reasons often that is not possible!
With very close subjects the required flash separation may only be a matter of a centimetres, however with distant animals it may be necessary to move the flash up to several meters away. The owl below was at least 20 metres away and the flash had to be moved more than two metres off camera to avoid red eyes.
When red-eye cannot be avoided by moving closer or moving the light source further away, a bright light can be used to shrink the pupils. The smaller the pupil, the less light it lets in and the less severe the red-eye appears as shown in the two images below. The downside is that the pupils are always contracted, which makes owls look abnormally intense.
Fixing red eyes
Sometimes you can't avoid red-eye and when this happens you can only remove it using photo-editing software. With very careful editing, the eyes can look quite natural. Pupils almost always have some light and colour in them, so I prefer to darken them by 'burning' the shadows and mid-tones, then desaturating until they are almost but not quite black. You can also 'dodge' the catchlights and other eye reflections to enhance the appearance of the eye.