Basically humans have one reflective layer and animals have two. Here is an explanation I found with a Google search:
"The consistently red color of the human reflex derives from the red blood pigment hemoglobin. Light from the flash picks up the red from blood vessels encountered during its bounce off the retina, just as reflected sunlight picks up the color of a red sweater.
Why, then, do animal reflexes come in so many other colors and seldom in red? The answer lies in the tapetum lucidum, a highly reflective, variably pigmented membrane backing the retina in animals with good night vision (including dogs, cats and most domestic animals) but entirely absent in humans. The tapetum lies directly behind the retinal photoreceptors. (Nova's The Nocturnal Eye nicely illustrates the anatomy).
The tapetum enhances low-light vision by giving retinal photoreceptors a 2nd crack at any incoming light that manages to escape absorption (detection) on the first pass. In dogs, at least, an additional boost may come from tapetal fluorescence, which shifts incoming wavelengths into better alignment with the peak spectral sensitivities of the photoreceptors. Tapetal pigments surely come into play here.
When tapetal pigment is present, its color dominates the color of a given animal's reflex. Tapetal color loosely follows coat color. For example, black coats and green reflexes tend to go together. Most dogs and cats show a blue reflex as their eyes mature in the first 6-8 months of life. Pigment-poor animals like blue point Siamese cats with no tapetal pigmentation show a red reflex for the same reason humans do."
Hope this is useful/helpful/just interesting!