Long-haired cats are particularly prone to matting. A mat is a clump of knotted fur that can cause a cat discomfort if allowed to mat down to the skin. Mats are usually caused by an owner's failure to groom regularly enough, so if your cat has hair prone to matting, comb it out at least once a week. If mats are allowed to get bad enough, the cat will likely need to be shaved, after which you can begin a regular grooming regimen.
Separate the mat. Identify a mat to work on and separate it from the surrounding hair.
Hold the mat at the base of the hair shaft. Grip the base of the mat with your fingers before you start combing to avoid pulling on your cat's skin.
Use the end tine of the comb to pull apart the mat. Gently pull and separate the mat, starting furthest from the skin and working down. If the mat is too big or too stubborn to separate this way, use a pair of blunt-tipped scissors and cut once into the mat, perpendicular to the skin. Treat the two cut sections of mat as two separate mats and continue trying to pull them apart with your comb. Always untangle a mat all the way down to the skin. If a little tangle is left over, it will quickly grow to a big one again, and it will pull on your cat's skin in the meantime.
Inspect and de-mat under the joints and behind the ears. These areas have the most friction during normal activity and are the most likely to mat.
Brush through the cat's coat with a soft metal-toothed brush. Once all the major mats have been pulled apart, brush through the cat's entire coat. These brushes will pull out any last bits of matted fur and should run though the coat smoothly if the cat is fully de-matted.
Cut off the most stubborn mats. If there are mats that are just too tangled to comb out, you can carefully cut them off using your blunt-tipped scissors. Be careful not to cut the skin, as the skin of cats is very thin and could bleed profusely if nicked. Don't worry about the appearance of a cut mat, since the hair will grow back quickly.