If you happen to agree that declawing a cat is cruel, you may at some point wish to clip the cat's claws. This provides the benefits of declawing (no scratching of things that aren't to be scratched is possible), while avoiding painful and permanent surgery (the claws grow back, just like clipped fingernails, and I have never seen a cat notice the absence of the points). For some cats (such as polydactyls), this process can be necessary to prevent painful problems with the claws. The first step of this procedure is determining which kind of cat you have: cooperative-cat, or demon-cat-from-hell.
If you have the first kind of cat, this will be a very easy process. Sit down behind the cat or place the cat on your lap, and pick up one of its paws. Press down on the pad on the bottom of the paw, and the claws should be forced out. Be careful to to press too hard or wrench the cat's leg, as this may put this cat into the second category. Using a pair of clippers (special claw-clippers exist, although I've heard of ordinary nail clippers being used), clip the point of the claw making sure to not puncture the blood vessel inside. This vessel should be visible as a dark cavity within the translucent claw, and will bleed profusely when punctured (so the vet claims; I have no intention of finding out). Repeat this procedure for all claws/paws that you wish to trim.
If you have a cat from the second category, prepare to deal with a sharp learning curve. The procedure is actually much the same as above, except with more care taken to avoid the untrimmed claws. The strategy that I find works best is to try and find the cat while it is calm and relaxed. Slowly approach the cat, and sit next to it (placing this kind of cat on your lap is not recommended, as it may decide to spay or neuter its owner). Petting this cat may help, or it may make things difficult, depending on the cat's personality and relationship with you. If the cat is likely to bolt, find some way to restrain it without harming it. I find placing an arm or a leg in front of him works on my cat. If the cat is very adamant about not having its claws trimmed, waiting is probably the best solution unless it's urgent. Remember: this cat still has its claws and knows how to use them.
Wuukiee says re How to clip a cat's claws: I recommend wrapping a hostile cat in a towel and pulling out the paws one at a time. Works wonders