According to animalplanet.com when cats are relaxed and happy, the muscles of the cat's larynx (also called the voice box) act as a valve for the air they breathe, causing them to vibrate as they breathe in and out. That's what makes the purring sound. It's a bit like the muscle reactions we have when we are happy: when a human is happy, we smile or laugh. Cat's don't have the same control of their face muscles as we do, but they have control of their voice box that we do not. Other creatures purr, too, such as bears, racoons, and badgers (for more animals that purr, see this article at Wikipedia)!
