It was long debated whether dogs could see colors or only black and white. Early research, and the popular consensus, was that dogs could differentiate brightness but not see color. Turns out, dogs can see color, just not exactly the way we do.
The eyes of both people and dogs contain special photoreceptor cells, or light catching cells, in the retina called cones. When these cells are stimulated they send a signal to the brain which is perceived as color.
People have three types of cones in our eyes allowing us to see a wide variety of hues. Dogs however only have two types of cones limiting their visual spectrum.
Dog color vision is fairly similar to a person who has red-green color blindness. Dogs can generally see a variety of shades of yellow, blue, and grey.
While humans have some advantages over dogs when it comes to seeing the world in full color, dogs are actually better than us when it comes to seeing movement and seeing in low-light. Dogs have more of a special type of cell in the eye called a rod. Rods are better at seeing in low-light and detecting movement than cones.
Also, in the dark, you'll notice that a dogs pupils will dilate and eyes will shine (almost as if they are glowing in the dark!). This shining is due to the tapetum lucidum (or "shining layer") which acts like a mirror, reflecting any light that enters the eye.