Most people believe that cats can't be trained.  Quite the contrary, cats can be trained quite easily if you apply the principles of learning.  Reinforcement is something that results in a behavior increasing in frequency.  Positive reinforcement in regards to animal training is often food however, toys and play can also be used.  The reinforcement is important information to the animal that it has done something right.  For cats, food may or may not be the best reinforcement. 

Some cats respond well to clicker training.  The first step is to associate the clicker sound with something desirable (such as food or play).  Once that is established, the click can mark a behavior when it happens, with the food to follow. 

Punishment doesn't generally work and is difficult to use because timing and intensity are essential to obtaining the desired result.  Punishment occurs after the behavior happens and does not give the animal any information on how further punishment could be avoided.  Punishment can also cause anxiety, fear, and aggression.  Cats (and other animals) respond best to positive reinforcement because they often are unlikey to associate a punishment with their actions.

Some owners may want their cat to "sit" on command.  Remember, a cat already knows how to sit:  what you want to do is put that action under verbal or stimulus control.  This is a perfect opportunity to use clicker training but first make the clicker relevant as decribed above.  Then, start by watching your cat or even calling him to you and waiting for him to sit down.  The instant that he does, you "click" and drop a food reward.  Then back up and wait again.  The cat is likely to approach, and when he sits, click and drop another reward.  As you repeat this scenario, the length of time it takes the cat to sit down decreases as he learns that sitting results in a reward.  The next step would be to add a word to this action.  As the cat approaches, just before he sits say the word "sit."  When he does sit, click and treat.  Soon he will learn that "sit" means put your bottom down and you will get a food treat.  Now all that is left is to phase out the food as described above.  Finally you can begin to link desired behaviors together.  For example, you call the cat, tell him to sit, and give him the reward.  you can use the same technique to teach the cat to jump onto a stool or through a hoop using targets, clicks, and rewards; the possibilities are endless.

Understanding how cats learn and their motivation for performing certain behaviors will help anyone begin to train a cat.  After all, aren't the lions who do all sorts of tricks much like our domestic cats?