There's no more impressive sight to witness in nature than the seemingly glowing, backlit eyes of a nocturnal or crepuscular mammal. Especially in the case of large species such as the leopard in the photo above, it can be an awe inspiring sight to behold. Not only mammals, however, exhibit the characteristic glow, or eyeshine, but a great many vertebrates. Actually this phenomenon is not a true glow, but rather a reflection of light.
The biological mechanism behind the characteristic nighttime eyeshine comes courtesy of a membrane situated directly posterior to the retina called the tapetum lucidum.
The highest level explanation of the mechanism behind the tapetum lucidum is thus: ambient light in otherwise dark conditions passing through the retina is reflected back to the retina by the tapetum lucidum membrane. This re-emission via reflection gives the photoreceptors, mainly rods, another chance to gather more photons for image processing.
A diagram is helpful in better understanding this process. In the right-most section of Figure 1 below, light enters the eye and first contacts the retina (red color coded). Light passing through the retina impacts the tapetum lucidum (green color) membrane and is reflected back to the retina. Thus the light sensitive rods get a second chance at absorbing and processing more light. Generally most vertebrates that have keen night vision have a greater proportion of more light sensitive rods than cones. Cones are responsible for sensitivity to color.
As always, any SMEs (subject matter experts) in biology/zoology feel free to chime in and correct any inaccuracies in my description. There is much detail involved in the biology of the tapetum lucidum; to get an idea of how granular the subject can become, take a glance at this article from the journal Marine Ecology.