Here’s a 2-minute video by SciShow that explains why the red-eye effect often turns irises into bright red spots in flash photos.
The basic idea is that when a camera flash hits an eyeball in a dark environment, some of the light often reflects off the blood vessels in the back of the eye and back to the camera before the iris muscles can contract.
Many cameras have a anti-red-eye mode that fires off quick flash bursts before the actual flash in order to cause the iris muscles to contract before the photo is snapped, thus reducing or eliminating the glowing red eyes.
Another interesting thing to know is that the red-eye effect can be used to check for eye problems — particularly eye cancer in young children. If a person exhibits the red-eye effect in only one eye and a bright white pupil in the other eye, they should have their eye examined by a doctor, as it could be a sign of retinoblastoma.