Cats are very territorial creatures. When a cat has lived with you for a while, it has staked out its "territory" in your home. You may notice that Kitty has a particular affinity for the couch or a particular spot on the bed. When you introduce a new cat to the house, both cats have to redefine the territory to claim their own spaces.
There is no foolproof way to convince multiple cats that they want to be friends, but there are several tips to make the transition easier and more peaceful.
When you bring home a new cat, start out by letting it explore one room with a door closed. You want to choose a room that you are in fairly often so that the new cat can get used to you. If the already established pets have favorite spots, you should probably avoid those so they don't feel like they are being kicked out in favor of the new Kitty.
New Kitty should stay in her room for a while, at least until after a vet has examined him and given him a clean bill of health. The animals will sniff at each other through the door, learning the new smells.
As the cats seem calmer about the new smells, try switching the locations of the cats. Let the established cats into the closed room while letting the new cat out into the rest of the house. This lets everyone see and smell everything and still not have to deal with any actual interactions.
The next step is to put New Kitty into a carrier or crate in the main part of the house. Let the other cats sniff at her and see her.
The final step is to let the cats all out in the main part of the house. There may still be some growling and swatting; don't worry, this is normal. The cats may challenge each other because they are trying to figure out who is dominant. The usual introductory cat fights are done with claws in and there is no damage done. The cats will eventually come to an understanding of one another. With any luck, the cats will become friends. But even if they don't become friends, they will most likely each find their own space and learn to live with each other peacefully.
Because cats vary so much in temperament, we can not tell you how long the introduction process will take. A very few cats will truly never get used to living with other cats. And there are some cats that don't need any introduction "process" at all - they'll happily play with any cat you introduce them to. The best advice we can give is to pay attention. If the cats seem to want to be with each other, then you don't have to worry much about violence and territory. If the cats are growling and fighting every time they see each other, take things slower and wait for the cats to calm down.
It should be noted that these introductory steps assume that all cats are spayed/neutered, have been vaccinated, have tested negative for feline leukemia and FIV, and are generally healthy. If any of these things are not the case, the cats should be kept separated until all of the animals are completely vet checked and healthy.
