Cats make certain sounds. They "purr" and they "meow". With meowing, the cats seem to draw attention to themselves for a certain reason. They may want to be petted or they may want food or milk to drink. However, cats purr for other reasons. For the most part, if a person went up to a cat and started petting it and it started purring, the person would feel that the cat is happy and wants to be petted at that moment.
However, purring can mean a bunch of different things. It can mean that the cat is happy, but it can also mean that the cat wants something. The cat may also feel hungry or be relieved. A cat may be healing from pain which causes it to purr.