It is commonly believed that cats cannot be trained. The independent reputation of cats makes some people love them all the more; others prefer dogs for the same reason. The fact is, as independent as cats are, they can in fact be trained to a greater extent than many pet owners realize. If your cat exhibits problem behavior, this should be a relief to you.
People sometimes think that where cats are concerned they either have to simply put up with a problem --whether it be aggression, destructive behavior or not using the litter box-- or get rid of the cat. This is, in most cases, simply not true. The vast majority of cats can be trained out of troublesome behavior.
Litter box problems are a common complaint among cat owners. When your cat does not use the litter box properly -or at all!- it is unpleasant, not to mention unsanitary. Cats are naturally clean, so using a litter box comes quite naturally to them. If a cat that was using a litter box suddenly does not, this is a definite warning of a problem. To rule out a medical problem, you should first bring your cat to the veterinarian.
If the cat has been medically cleared, make sure that the litter box is kept clean. Also, the litter box should be kept in a quiet area, and not right next to the cat's food and water. In this respect, cats are like people; they like a separate "bathroom" that has a certain amount of privacy. The box should also be the right size; big enough to accommodate the cat comfortably, but not too big.
Do you have an aggressive cat that claws at you or bites? This can be a nuisance, as well as painful. Fortunately, in most cases cats can be trained out of such behavior. Once again, if this is new behavior on your cat's part, have it checked out medically. If this is a long-term problem, you may have a cat who is not well socialized, or even feral. For such cases, you can usually tame the cat, but it takes some extra time and patience.
If your cat is aggressive, try to become sensitive to its moods and body language. That way you won't try to touch him/her at the wrong time. Also, make sure your cat gets enough exercise. Take some time every day to play with him/her, but not too roughly (especially if aggression is the problem!). Playing has the dual purpose of giving your cat exercise and socializing him/her. Make sure your cat is spayed; this will, in general, make for a better pet and will curb aggression.
I have provided a starting point in addressing some common behavioral problems. You may need to further educate yourself and spend some time working with your cat. Just remember that the vast majority of cats will respond to the right kind of attention and can be transformed into wonderful pets!