Why do cats purr?

According to veterinarian Bruce Fogle, author of The Cat's Mind, the original function of purring was to enable a kitten to communicate with his mother that things are well. A kitten is able to purr by the second day of life, and although he can't meow and nurse at the same time, he can purr and nurse. And the mother cat often purrs back, probably to reassure the kitty.

Source: http://www.petplace.com/cats/why-do-cats-purr/page1.aspx

cats purr to show that they apreciat what you are doing.

cats purr when they are content and quite relaxed

There are several reasons for this. For one, they use purring to communicate with their mother when they are still very small. However, they also do purr when they are happy, content and relaxed. Lastly, a cat's purring is sending vibrations through its' body which supposedly help them heal little injuries faster. This means they might also purr when they are going through a process of healing. By the way, a purring cat on one's limbs can also help the healing process of the human bodies - and it is cozy at the same time.