How does one properly write a combat scene?
I don't think there's any one answer to this, it would depend on the tone and style and how important it is to the main character.
If it is just two people introduced a paragraph ago, readers are less likely to care
I personally avoid them. Unlike a movie, it's hard to make fight scenes add to the story. Everyone here have good advice, but I would really ask, "What is this scene adding to my story." Also, keeping it short and fast paced might be a good idea. There isn't a lot of ways to make it exciting. A two page fight scene can't be anything but boring. A paragraph probably can be exciting.
I agree with Mizal that there is no single way. What I am good at is a gritty style, so I will answer from that perspective. For me what makes combat scenes fly is that there are different stages that happen in very quick succession with dramatic changes tempo and voice.
I plot these phases out beforehand. Example: The dread before the fight, the tension as opponents are waiting for each other to move, a flurry of blows being exchanged, a desperate scramble as things go south, a desperate counterattack, a moment of calm as combatants circle each other catching their breath...
You want to reflect the very different nature of these phases in the language. In the fast parts, sentences are simple. Breathless. Cut short, by a lack of breath. Fragments are acceptable.
Longer sentences prevail in the slower parts, where opponents take a step back and assess their wounds, while keeping their eyes trained on the enemy, who is stalking just feet away. Writing these long, slow rolling sentences, with excessive detail, where you see the individual beads of sweat running down the enemy's face, gives your writing a dreamlike, yet highly-focused, tone.
Combat is primal and to drive this message home you want to keep your word choices well grounded. Go for words with Germanic or Celtic roots and weed out words with French or Latin roots. In combat there is no intestine, it's guts, etc.
Another characteristic of experiencing combat is that the brain chooses to distribute its attention in odd ways. In the fast bits we might not have time to notice where exactly the opponent is standing or what happens around us, but at the same time we perceive some minor details in absurd larger-than-life clarity. The reader might get a description individual bullets streaking through the air or the protagonist might remember that these are 7.62mm cadmium-tipped high velocity rounds.
One of the big challenges is that movies have trained us to expect visual surprise in combat. Writing just doesn't work the same way. If you were to transcribe combat scenes from a Jackie Chan movie, so much explanations would be necessary that the pace would be destroyed. You would literally end up with something that reads like a technical document. One way around this is to avoid the special effects altogether and stick to simple moves that the readers can easily picture, and hope that their brains will do the rest. (Or to echo @Zake , in writing we can have psychological depth instead which can be much more interesting than visual theatrics.)
In my opinion one can usually get away with some fancy bits in a combat scene, just not too much. To make this work it is essential that you and the reader are on the same page when it comes to the fast bits. The cheap way to do this is to rely on genre cliché. In a martial arts piece I can write, 'He jumped right back on his feet,' and basically trust that the readers have watched enough Kung Fu movies that they know how this looks like.
Another tool is to establish a shared vocabulary beforehand: Books that feature eastern-style martial arts combat often describe extensive training sessions in great detail. Apart from building setting and character the main reason for these is to establish the vocabulary with the reader. When it comes to real combat you can write, 'He jumped right back to his feet, took two quick steps and launched into the whirling dragon kick.' Because we have established beforehand what such a kick looks like we don't have to slow the pace with a cumbersome description, so instead our reader has time to think 'Oh shit, he is doing that!'
If you go for large battles, nevertheless keep it personal, but throw in short statements about the bigger picture. The classic here is of course the pirate battle, where the captain's duel while the larger fight rages around them (and reflects every move in their personal encounter).
Finally don't forget about the environment. In the very least we want to trash the furniture, but few things are as much fun as a sword fight in a burning building, where missed blows send sparks flying, and burning bits of ceiling rain down. Again the rule is we don't want to bother the reader's brain with new stuff while the combat is fought, so all the objects from the environment that you use should have been at least mentioned before you get to the action starts. Of course, some things go without saying. In a Saloon fight, grabbing a bar stool won't raise eyebrows, but if the protagonist gets thrown into a piano, you better mention the piano before the fight starts.
I think Ninja is right and Muay Thai was definitely helpful in that I have a pretty clear picture of how people are standing and how they move, but I think what was more helpful is that I just really like movies and try to think carefully about what makes my favourite movie fight scenes. In general, I think, you need to have a clear progression from the beginning to the end; like, the circumstances need to change, if that makes sense? People get injured or tired, someone becomes enraged and fights twice as hard, shit like that. The audience will tune out if there isn't some kind of dynamism at play.
I try also to maintain a sense of "dramatic rhythm", as well. One person acts, another responds. If one guy just slashed with his sword, I won't have him speak that line. This isn't a hard and fast rule, but I kind of like to trade the action off and I find it works better.
Also, keeping it brief is essential. Once you're beyond like three hundred words of battle with no dialogue or narrative, you're losing the audience, I think.
Honestly, reading Warhamer 40k novels when I was younger I think helped develop my love of writing combat because those books are like 80% violence and some of the are fucking gripping.
Also, edit for concision always. If you can say it in fewer words, it's generally better.
I feel like what precedes the combat, and the peripheral events that happen during the combat, may be more important than the combat itself. Describing the action is necessary to an extent, but if it's constant description of movements your combat will get boring quickly. Where do you draw the line between technical action, dialogue, and description of emotion and other things not directly related to the fight itself?
It's a balance I'm still trying to find. I think my weakness is a tendency for too much exposition, but I do try to add elements that tell a miniature story as well. A couple years ago I felt like I was getting close in a storygame I predictably never finished. This is what I had written then in a rough draft (if this helps you, great; if you find something that needs improvement, by all means please comment, but the point is to show what I think is one decently done thing and one poorly done thing, which I will specify at the end):
Things were not going well. [insert stuff about how intelligence was wrong and the lines have been breached and repaired numerous times throughout the day]
During the last twenty minutes, a lull in the combat nearby had allowed you to catch your breath, and despite your instincts protesting against it, you rested on your knees, using your sword and shield to support you.
A shrill voice from behind you draws you back to attentiveness and you stand. A boy runs up to you, carrying a skin of water and parcel of some sort. "Captain, General Mathers sends word that the 43rd Division will be unable to aid the Northeastern flank." The boy, hands on his knees, takes a deep breath and continues. "They were attacked from behind by an auxiliary force of mercenaries, and while disposing of them another hostile unit from the north arrived, this one bringing more than men."
Troubled by this new development, you watch as the boy takes a drink from the skin and then hands it to you, his breath still coming out in ragged pants. He can't be older than 1o or 11. Sparing a quick glance to your left to make sure the line is holding steadily, you ask, "What do you mean by 'more than men,' son?" Turning back to the melee, you take a deep drink and await the answer. In the distance some twenty more enemy soldiers approach.
"Ogres, Captain," says the child. "General Mathers says at least ten accompany another squadron of enemy soldiers, and that they'll have to make their retreat this evening. They were nearly routed when I first ran, Captain. He begs you consider sending the running reinforcements here to the East to help along the Ilowaen Forest, and begin your retreat one night early to meet him at the next defensive position."
You feel frustration bordering on panic at that. Ogres? Rumors were already circulating among the commanders nearest your position about reanimated dead soldiers attacking on some portions of the Eastern perimeter, and what little you knew of the overall defense plan and intelligence suggested a powerful magic user may be behind the attack. But ogres? Twelve feet tall and nearly half as wide, alone they were terrible foes. If the enemy has somehow convinced them to join them in numbers against the Kingdom, things are far dire than you had previously believed.
A flurry of shouts and metallic clangs to your left signal another breach along the line. Resignedly you raise your sword and shield. "Boy, what's your name?" you ask, eyes never leaving the fighting just ahead. One of your men gets run through with a spear, and then two enemy soldiers spot you and begin to approach at a fast walk.
From behind you the boy says, "Oliver, if it please you, sir." Not taking your eyes off the two approaching men, you loosen your shield buckle and let it slide down.
"Grab the shield, Oliver, and run back twenty paces. Keep the shield up in case that one throws his spear at you. If I fall, strap it to your back and run as fast as you can to my second in command, Lieutenant Marks, and inform him that he has command. Deliver him the message from General Mathers and then retreat to the next position." You look back and see the boy frozen, staring at the shield. "Oliver!" you yell. "Do you hear me, boy?"
Shaken back to awareness, the boy nods. "Then go, Oliver, now!" He picks up the shield and begins to run back. Knowing the boy is temporarily safe, you turn in time to see the two soldiers reach just outside of your attack range, their pace more deliberate now.
The two fan out as they approach, flanking either side of you, and in the distance you see a third approaching. The one on your left holds a spear, the other two short swords and shields. The flanking soldiers exchange glances, and then the one to your right screams and charges. Anticipating the charge is really a feint to draw your attention away from the spear wielder, you immediately turn to your left, swinging your sword down. As you expected, the other man planned to stab you from behind, and your sword deflects his spear to the ground.
Stomping hard on the blade, you pin it to the ground with your foot and sword, simultaneously drawing your dagger from your belt with your left hand. In a single, swift motion, you fling the blade underhanded at the spear wielding man, and the dagger strikes him directly in his unprotected face. You turn back just in time to deflect the sword strike from the other man with your own sword, and then kick his shield to gain distance. As he stumbles, you turn back to the other man and finish him with a stroke to the neck from your sword.
Pulling out your dagger just in time, you use it to parry an overhead swing, pushing it to your right, then you step around the man and stab your sword into his right side, just underneath his armpit. You quickly slash him in the throat with your dagger and he falls, hot blood soaking your forearm.
You pivot to face the final enemy, but before you can react, a savage strike lands on your unprotected upper left arm, exactly where a hole in your mail armor had been made earlier in the day. The pain is shocking, but you manage to hold on to your dagger. The enemy steps back, and the two of you square off.
He slowly swings his sword in an arcing figure-eight, and then leaps forward, attacking you on your left side, trying to gain an advantage from your wound. Knowing your left arm may be too damaged to go where you want it to, you jump back and parry with your sword instead, taking the defensive. You try to raise your left arm back up to use the dagger as a fighter would with a rapier, to parry attacks and thrust with your sword, but the pain nearly makes you feint. You let your left arm hang low, but hold onto the dagger so your opponent is forced to assume it is still a viable threat.
The man attacks again, and you are again forced to step back. Suddenly your foot lands upon one the men you just killed and you slip to the floor. The man comes in for the kill, but a blow from behind causes him to turn back: the boy named Oliver struck him with the shield. Wasting no time, you quickly leap to your knees and lunge at the man's groin with your sword. In defense he swings down at you, but you are able to raise your dagger arm high enough to take the blow with the flat of the blade, holding it reversed so your forearm supports it. You drive the sword in deeper, and the man's own sword falls from his hands. Standing, your sword still deep with him, you cut his throat with your dagger.
As the man falls to the ground, you look at Oliver. "I would chastise you for not following orders, Oliver, but you saved my life. Thank you." The boy, still in shock, doesn't seem to hear you. His chest heaves rapidly, and he sways before you. You slip your dagger back in your belt and place your hand on the boy's shoulder. Kneeling, you say, "Oliver, look at me. You're okay. You're a hero, son, and I need you to carry my message." The boy seems to perk up at that and meets your eyes. You hold his gaze for a moment, then you squeeze his shoulder and stand.
Now, what I like about this excerp is the little side character Oliver gets a tiny bit of interaction with the player, and manages to go through his own mini-story arc. It's a little bit god-from-the-machine, but I feel like it adds some additional depth to the battle situation. I feel that too often, combat scenarios, especially battle sequences, forget about the peripheral things that go on, like messengers running from position to position to deliver intelligence when other means fail. I feel like here, that was done pretty well.
What I don't like is I feel I'm a bit too descriptive with the combat. I think I can streamline it, maybe a lot. Some of it I think is necessary and adds some character (like the main character having an injured left arm, but refusing to drop the dagger so that the enemy must respect it as a threat), but some of it was way overboard in move-by-move description.
So, the point of this example was to show what I believe is one thing done passingly and one thing done poorly in combat description. DO add some things not directly related to combat, some stuff on the periphery; DO NOT spend an excessive amount of words saying which sword goes where and how long which swing lasts. Give enough information to paint a reasonably clear picture, but save the detailed description only for the key moments.