Why do cats fight?
Cats can be very territorial. I have found (from personal experience) that males especially are prone to fighting.
I have had multiple cats in our household and some males will be alpha, some males more laid back (omega).
It is also possible for females to fight and males and females to fight. It really comes down to the cats and their personality. My experience has been that of male to male fighting. Some households will never experience fighting at all.
Cats who roam outdoors are much more likely to become involved in cat fights. Either by wandering into a neighbouring patch which has another cat or a cat encroaching on your cat's territory. We've all heard the caterwaul late at night.
There are a number of risks your cat is exposed to when he engages in fights. Not just injury from bite wounds and scratches, but he also becomes exposed to cat diseases such as Feline Leukemia and Feline immunodeficiency virus (FIV/cat AIDS).
How to break up a cat fight (safely):
Never, ever get between two cats having a cat fight. You risk severe injury in doing so. Cat bites and scratches frequently become infected and it is just not worth the risk. If you do need to break up a cat fight the safest way to do so is throw a blanket or towel over the cats or use a broom to separate them. Be aware that cats will be very wound up after a fight and you are still at risk of being bitten or scratched after the event. Give your cat(s) time to calm down before approaching/petting them.
How to reduce the risk of cats fighting:
Desexing all cats goes a long way to reducing fighting among cats. Entire cats are considerably more territorial. That's not to say that desexed cats don't fight at all, they do, but desexing cats will certainly reduce the incidence of fighting.
Don't have too many cats. The more cats you have, the greater the chances of fighting among cats.
If you have one cat and plan to get a second, seriously consider getting one of the opposite genders. If you already have an alpha cat, it is easier to introduce a kitten to the house than an adult cat as younger cats will be less of a threat to the already established cat in the household.
Discourage neighbourhood cats from coming onto your property.
Keep your cat(s) indoors or in an enclosure, which reduces their exposure to other cats from the area.
Fighting among household cats:
This is a common problem and a great cause of concern when it occurs. I have had multiple cats over the past 20 years, most of whom get along just fine, but sometimes you will have cats who just don't like each other. I have found this to be the case far more often with male cats (always desexed/neutered). They both fight for the top "alpha" spot.
There are ways to reduce this kind of behaviour occurring.
Firstly, make sure your cats have enough space. Are there enough litter trays, beds, cat trees, different feeding stations? It is important to provide enough of the above so that your cats don't have to share if they don't want to. Each cat should have his own space.
Try Feliway diffusers, these contain synthetic pheromones which are calming to cats.
If you have two cats in particular who just can't get along, you may need to separate them and re-introduce them, slowly. To to this, each cat will need to be confined to a separate room for a period of days. Keep their water bowls and bed with them. After several days, swap cats over so that cat a is in cat b's room, cat b is in cat a's room. Take a blanket or item of bedding from each room and wipe it down on the other cat. So, take a blanket from cat a's room and wipe it over cat b, a blanket from cat b's room and wipe it over cat a. This is to transfer scents between cats. Do this for a few days. Now it's time to re-introduce. Bring the cats out, in cat carriers. Leave them for a few minutes. Let each cat nibble on a treat while he is in his carrier. Slowly open the carrier doors and let the cats come out. Now is the time to be diligent. Talk calmly to your cats, let them explore, sniff the area. Continue to give the cats small treats and reassure them. If a fight erupts, return the cats to their separate rooms (remember, never physically get between two fighting cats, you will get injured), and try again in a few days. After a period of 30 minutes, return the cats to their respective rooms. Continue the re-introduction over a period of days, gradually extending the amount of time the cats are out.
If you have one cat who is especially aggressive/dominant over other cats and the above methods haven't helped, it is a good idea to speak to your veterinarian who may recommend drug therapy for your cat.
In some cases where despite all your attempts, the cats still continue to fight, it may be time to consider rehoming one of them. I know many will say that a cat is for life, but sometimes two cats just can't get along. We had this situation several years ago. Two very alpha males in the household, who incidentally left all the other cats alone (we had 5 at the time), but they fought and sprayed constantly. One cat was also aggressive to humans, so rehoming him was out of the question. The other cat, Nicholas, a gentle natured boy went to live with a good friend of ours where he was an only cat. Both cats stopped spraying when they no longer lived in the same house. Rehoming was not an easy choice to make, Nicholas was a one in a million cat, but it was the best choice in a difficult situation. He was happier on his own and our other boy was happier without a second alpha male to stir things up.