Red eye happens when a large burst of light is reflected off of the blood vessels in the retina- thus the red color. You may have noticed that animals have a similar glare in their eyes, but maybe it is not red in color. That is because the light is reflecting off of a different part of their eyes that helps with night vision called the tapetum lucidum. Interestingly, if you notice that the reflection in one or both of your eyes is discolored consistently in your photos - a yellow glare instead of red - it may be an indication that you need to have your eyes checked.
So, why precisely does this happen in some photos, but not all photos? There are three main conditions that need to be met for red eye to occur: the angle of the eye and camera, the size of the pupil, and the lighting of the room.
The angle of the eye is important because it must be such that it both catches and reflects the light back and the camera also must be in a position to capture the reflection in the eye. This is why the effect usually happens with amateur photos and non professional camera equipment. If the flash is coming from the camera and the eyes are looking directly at the lens, then the situation is perfect for red eye to occur.
The size of the pupilmatters for a few reasons. Pupils being large will allow for more light to enter the eye and reflect off of the retina. Children have wider pupils and are more likely to have red eye in amateur photos. People who have been drinking also have slower reaction times and therefore may have larger pupils at the time the photo is snapped as their eyes will not react and adjust to the flash quickly enough. With wider pupils the eye can capture more reflection and have a wider angle for the reflection to be caught on camera.
Finally, lighting. As photography is very dependent on lighting, ambient light and the setup of your flash will also determine the likelihood of red eye occurring. Low ambient lighting will make your subject's pupils dilate and therefore susceptible to reflections when a flash happens. Some cameras have a setting to do a pre-flash to help the pupils close and reduce chances of red eye. Finally, if you can set your flash up somewhere instead of directly in front of the subjects, they won't have light beaming directly into their eyes. This will help keep the pupil from dilating and change the angle of the light and reflection in the eye.
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More reading:
https://www.researchgate.net/publication/228649986_Automatic_Red-Eye_Removal_for_Digital_Photography