We should be thankful to the technology that has simplified the art of taking pictures, but the effect of red eyes still ruins our beautiful pictures sometimes. What happens is that in low ambient light, pupils enlarge to allow more light to enter the eye. Thus, when the light from the flash enters your eyes, the pupils do not have enough time to constrict. Therefore, the amount of light entering your eyes reflects off the red blood vessels of the choroid, a layer of connective tissue at the back of the eye. When the flash of a camera lights up the blood-rich retina, it results in the red-eye effect.
So the reason you have red eye is probably that you stare directly at the lens of the camera. You can simply remove the red-eye effect by simply looking at a nearby light bulb just before taking the photo.