To make it simple, cats and dogs can see some colors, but not all. Cats and dogs have deuteranopia, commonly called red-green color-blindness.
The inset picture in the image above shows what a cat might see when looking at the larger picture. To sum it up, cats see blue and yellow fine, but other colors are problematic.
The image was created in Gimp using the colorblind filter and a screen capture. To use the filter, click View → Display Filters… and select what type of colorblindness you want to check.
Veterinary Topics: Feline Vision
The Straight Dope: Are cats and dogs really color-blind? How do they know?
Colorblind Web Page Filter
The Island of the Colorblind by Dr. Oliver Sacks.