Most of us are bound to have some kind of scrap or another turn up in our stories. For some perverse reason, writers like giving their characters a scar or two (it builds character!), and readers like holding their breath, waiting for the outcome of the action. "There need to be more battles!" my father exhorts.
I'm not going to go into the psychology (or psychosis) behind having fights in novels. Whether or not you do have fights is dictated by circumstances and the nature of the characters. On the one hand you might have, say, a Bjorn Bjornson from Rosemary Sutcliff's The Shield Ring, who slugs his opponent dead in the face for calling him a coward. In his culture, calling a fellow a coward had much more serious connotations than it necessarily does today, and the reader, picking up on that, feels a thrill as Erland goes down with a bloodied face. On the other hand, you might have a Beowulf, a man from the same kind of culture, politely passing Unferth's backhanded insult off as merely the effects of wine so that (though he knew he could beat Unferth in a heartbeat) he would not cause a stir in his father-friend's hall. The reader, picking up on that magnanimous gesture, cheers for Beowulf.
The insults, the culture, even the men (though very different in age) are very similar in both stories. So what decided that the one should fight, and the other not? What makes the one outcome as believable as the other? I'm going to sound like a broken record: know your characters. If you were to set two characters, very similar to each other, in similar situations, with similar provocation, could you be sure that they would not react perfectly alike?
The problem is, I have found, that the writer is too often caught up in the action with the character. This sort of catching up is all very well and good for a reader, because that's the purpose. But when you're trying to write a decent squabble, you can't let the blurry red of a fight get between you and the black ink on the page. Take a moment to find yourself before you plunge in, if you have to. On the other side of the coin (heads and ships, if it's Roman) if the writer is able to feel that wild buzz of excitement as the words spin off the pen, or the keys rattle beneath the fingers, chances are the reader will be able to feel it too. But the blurry red should probably take a back bench - that sort of mindset isn't conducive for clear, level-headed choreographing.
After that, put on the character's boots. Even if the whole fight is being narrated from a third party's view, getting into the place and mind of the person potentially throwing the punches helps dictate the outcome of the scene. Again, it all comes back to knowing your character. If I were writing Bjorn, I would be stalking along with a grey sick feeling in my stomach, and the annoying blurry scarlet of fury beginning to swallow up my vision until all I could see was Erland's face...but the whole section is told from Freya's view, a third person altogether. If I were writing Beowulf (that sounds horribly presumptuous), I might feel that giddy golden uprush of mead and mead-smell mingled with the firewood scent, and stand a little taller over the others than usual, smiling gently down at Unferth in a knowing sort of way, a laughing, knowing sort of way, and brush off the man's words as a man might brush off a bit of plover-down from his sleeve...though the whole scene is told as from a third person's view.
There are scores of ways to fight, whether physically or verbally or covertly. Keep in mind that a man usually has a good idea of how to swing a fist, whereas a woman typically flings herself into the fray like a cat. A man can crush with his words, but a woman can nag and chip and wear and grind (or scream like the devil stepped on Tinker's tail). A man might stab you in the back, but they say poison is a woman's weapon. These are typical situations. Please, please, please don't have your characters use martial arts if they have never learned it. In the film "Serenity" good-guy Malcolm Reynolds goes up against the sinister Operative...and gets sorely beaten. A good story-teller won't suddenly give Malcolm fantastic skills with which to at least match the highly-trained Operative; no, Malcolm's bar-room knuckle-sandwiches, appropriate for the unsteady brawling of back-moon saloons, can't compete with the honed precision of the Operative's offensive moves. Keep it real - but also, if you don't intend to have your character die, give him a way of escape: it was the woman Inara who put black powder in the incense and effectively caused a diversion to get away.
What tips the balance the other way? What smooths the hackles - if the hackles are smoothed at all? Two characters of mine, always at each other's throats, never actually come to blows. The one is far too small (and knows it) to stand a chance, and continues to not only prompt but diffuse the fury in his opponent. On a basic level, they don't use the same weapons and can't quite meet in a fight: the one is used to employing his physical attributes, the other has always relied on his wits. Though a fight is always pending, it can never quite break: it has no outlet. But people are not static: keep in mind that such a storm must break somehow: your characters as well as your plot will determine which way the two characters will fall out: either together or apart.
If your characters fight, how do they fight? Each different character is bound to fight a little differently: if fighting is a key point of your story (a real to-the-death fight, not just a scrap of fists) you'll probably want to know the answer to this. What sort of weapon would your character's history and lifestyle have made familiar to him? In a story, you have the opportunity to choose which weapon best suits your character's personality. Keep in mind that bows take not only practice to fire accurately, but time to build up the appropriate muscles to draw. A sword is a mighty hunk of metal to be swinging around all day: even a hero's arm is bound to get tired. My fairy is unlikely to snatch up a shield and an axe and hope to have any luck with them; my big brute Catti is unlikely to pick up a bow (girlish weapons, his people call them, but perhaps only because they could never make a decent compact one) except perhaps to beat someone's brains out with it. My fairy, if inclined to fight, fights like a dance: light on his feet and quick; my Catti, when he is fighting, fights like thunder: with a bang and a boom that will knock you right over. People are of such varying temperaments and sizes, they will all come at a brawl a little differently. Keep this in mind.
The peacemaker is probably the most difficult character to write. By peacemaker, I don't mean "It's not going to help, having a row between you two." I mean someone with the guts and grit and wisdom and wherewithal to wrestle an explosive situation into tranquility. My mother used to tell us, "Do what makes for peace," and I have never forgotten that. It isn't a passive resistance to be irritated, it's an active and benevolent force. Sometimes it has to be a very strong force: the airforce didn't create the intercontinental missile "The Peacekeeper" for nothing. The peacemaker, whether by physical exertion, by rhetoric, or by simple presence, is the person who keeps the tapestry from snarling as it's woven together. These are such characters as Bilbo (The Hobbit), Muggles (The Gammage Cup), and Hwin (The Horse and His Boy): strong, gentle characters with surprisingly well-tempered backbone and courage. They are, I think, the most difficult character to write, but the most rewarding as well.
My father once told my brother (years ago, when we were all much younger than we are now, and the earth was young and the sun was hotter, and the light out of Elvenhome could be seen on the western horizon on Midsummer's Eve), "I don't care if you end a fight, but I don't ever want to hear that you started one." Know how to pick your fights, and how to fight them when you do - and, furthermore, learn how to diffuse them with dexterity.