INTRODUCING A NEW CAT.
Cats do not necessarily want a new friend. If you just put a new cat in your home, the resident cat may well attack it. Once a fight has broken out, cats will dislike each other and it is difficult to change their attitude.
Your responsibility should be primarily for the cat or cats already in your care. Think hard before asking a very elderly cat to put up with a new companion. Kittens will be accepted better than an adult cat, but kittens often pounce on and pester old cats. Some old cats will mother a kitten; others definitely won't!
THE INTRODUCTION SHOULD BE SLOW
Give both cats their separate core territory. Your aim is not to make these cats become friends: merely to avoid aggression until they have learned to tolerate each other. Let the new cat out in its own room. If possible add a retreat place like a covered cardboard box with entrance hole, a roofed cat bed, or a crate. Put familiar toys (if available), feeding bowls etc. in the room. Put litter tray in the room. This has got to be a safe place for the new cat. Let the resident cat sniff under the door of the room where the new cat is installed.
Mix scents. From the point of view of a cat, anything which smells strange is an intruder. Label cloths with the name of each cat, keep in separate plastic bags, wipe the cloth on the named cat, then present it to the other cat, wrapped round your hand. Let the other cat investigate this but do not force contact. Now mix the scent of the cloths. Wipe these on the resident cat and the new cat, then rub this in the resident cat's territory and in the new cat's room at chin height.
If each cats seems to accept the smell of the other, then give the new cat something to sleep on which smells of the old cat and visa versa. Keep swapping bedding. Transfer a little used litter from the new cat tray to the old cat's tray and visa versa to mix smells.
While the resident cat is shut away, let the new cat explore the house. Next, let them see each other by putting wire mesh across the doorway to the new cat's room or see each other through a glass door. You can buy a simple door hook and fix it so that the door is held very slightly open with the hook. (If you are good at DIY you can make your own screen door. It involves making a rectangle frame, adding insert hinges to the screen door and the original door frame, then fixings to the screen so that the screen door can be secured).
A kitten can be introduced to a crate over a few days (with food and water), then the crate can be placed where the resident cat can see it. If you don't have a crate, you can place a kitten in a cat carrier for this visual introduction.
If there is a lot of growling and hissing at this stage, or if they try to fight through the bars/mesh, you may have to reconsider whether you should add another cat. If they remain calm, give treats/food to reward them but leave plenty of space between the treats given to cat A and those given cat B. Play games with each cat. These visual introductions should be short and frequent. Both cats should be able to retreat.
If the cats remain calm, then they can have short periods of being in the same room - always supervised. If it looks as if there is going to be fighting separate them before this happens. As soon as they are calm, then supervision may cease. Both cats should have their own core areas so keep the litter tray, food bowl etc in the newcomer's room. The aim is not to get the two cats living close to each other, merely to let them easily avoid each other. A Feliway Friends diffuser can help this process.
Aim at allowing them to live separate lives, in the same house for as long as they want to. For two cats make sure that there are at least two feeding locations - cats do not enjoy feeding close to each other though it looks cute to us. There should be two litter tray locations. There should be several cat beds or cat resting places. Each cat should be able to eat and eliminate without having to worry about the other cat ambushing it.
Accept that that the new cat and the old cat may never be friends. If they tolerate each other, that should be good enough. But if one cat is seriously bullied -- not able to eat without being hassled, ambushed on its way in and out of the litter tray, harassed in its cat bed, spending all its time under the bed, then think of rehoming.