Brushing
Your cat needs to be brushed frequently to untangle hair and remove dead hairs.  Short-haired cats should be brushed one or twice a week, while long-haired cats could require daily brushing (this will depend on the cat).

You should always brush going in the direction of your cat's hair growth to avoid irritation.  I've noticed that a lot of excess hair collects right above my cats' tails on their backside, right above their tail, so in this area I will brush going against the fur to grab out dead hairs first, and then resume my normal movement.

If your cat gets tangles, hold the clump of fur closely and firmly next to the skin.  With a flea comb (pictured below), start at the ends of the fur and brush away from your cat, moving more into the tangle with each stroke.

As with all pet supplies, there are numerous brushes and combs available.  Most range between $3.00-$10.00.  You should note that the softer the brush bristles, the less effective it will be in removing dead hairs and tangles from your cat's coat, even though it will result in a nice shine.  To remove dead hairs, you should look for a brush with long pin teeth.  Below are two brushes that I highly recommend both, both sold at most pet stores.

Easy Groom Cat Soft Slicker Price:  $6.99 at Petsmart This brush contains long pins that massage your cat's skin while sorting through his fur, removing dead hairs.  My cats like the way it feels and will even turn over on their own mid-brush so that I can get both sides.  This brush leaves their fur feeling silky soft, looking shiny, and feeling wonderful.  Large balls of hair are easily removed from the pins after grooming.

Easy Groom Cat Flea Comb Price:  $5.99 at Petsmart Designed to remove fleas from your cat's coat, this comb is also great for removing tangles.  It is also great for getting into smaller areas, such as around legs.  Although hair needs to be removed from the comb a few times during grooming (depending on hair length), it is as effective as the Soft Slicker for removing dead hairs.

Your local pet store will carry cat bathing products, and you can even use baby shampoo (make sure it's tearless) which has been said to reduce dander to alleviate human allergies.  There is also wipes and waterless shampoos (which comes in a powder form) that you can use to clean your cat.  I have never used these so I cannot offer any advice (I prefer to wash my cats in the traditional way), but this may be an alternative you'd like to explore.

Some cats like water while some don't.  If you have a feline that fits into the latter category, you've got a job on your hands.  It if recommended that you find a friend to help you give the bath - one of you holds the kitty down gently in the water (a few inches in the bathtub works best since you have leverage over the frantic feline) while the other pours water with a cup and washes with the shampoo.  Do this gently but quickly and talk to your cat, telling him how good he is doing.  Make sure you rinse all of the soap out of his fur - you will end up with a kitty with irritated skin if you don't.

Make sure when you are done you dry your kitty off as best as you can with a towel (once again, don't go against the hair growth since this can be uncomfortable for your cat).  You can even put a hair dryer on low to make the process go faster if the noise doesn't bother your cat.  When you've done what you can, let him go and be on his own for a while.  Catch him later and finish toweling him off.