This article is from the General Cat Care FAQ, posted to rec.pets.cats newsgroup. Maintained by Cindy Tittle Moore with numerous contributions by others.
Start early with your cat. The younger it is when you begin grooming it, the more pleasant grooming will be for it. A cat that fights grooming may need sedation and shaving at the vets for matted fur; it is well worth the time to get your cat to at least tolerate grooming. Start with short sessions. Stick to areas that it seems to enjoy (often the top of the head and around the neck) first, and work your way out bit by bit. Experiment a bit (and talk with your vet) to find the brush and routine that seems to work best with your cat. Even short-hair cats benefit from grooming: they still shed a surprising amount of hair despite its length.
Thick, long fur
Inexpensive pin-type (not the "slicker" type) dog brushes work well. You may choose to followup with a metal comb; if you use a flea comb, you will also detect any fleas your cat may have.
Silky long fur
Soft bristle brushes work well.
Try an all-rubber brush, often sold as kitten or puppy brushes.