Here's a great video on why our eyes shine red in photos:
Once you have that down, then you can think about the tapetum lucidum, which is an extra reflective layer behind the retina in dogs, cats, and other animals with better night vision than us. That layer in most dogs causes a different color glow in dogs' eyes (sometimes green) when they are looking directly at the lens when you take a photo of them. The tapetum changes over the course of an animal's life, causing the color to be different in animals of different ages when you see this effect. Also, some breeds of cats and dogs have tapeta that differ in ways (like pigment) that cause them to shine different colors, including red, which often happens when the tapetum lacks enough pigment to block the reflection of light off the blood vessels in the retina.