I think we've talked about this a few times before, but when it comes down to it, at its core, combat in table top RPGs can be downright boring. Compare your normal TableTop fight to the scenes you see in the moving, no comparison right? I mean, you may occasionally get a good mental image for things, but ultimately you're throwing dice, rattling off numbers and it is all very mundane and mechanical. Not very fun, and very mentally taxing while not being fun. So lets see what we can do to fix that.
Step 1: Loosen Up
The first step in doing this is to loosen up. Lighten up on the mechanics of the fight, let people do things a bit easier, a bit cheaper. Let a player say they hit the guy in the head if he really wants to, maybe don't give the mechanical benefits a head shot would give, but the important thing here is to engage the mind not the mechanics. So loosen up a bit on those rules. If someone wants to do something that technically isn't allowed by the rules, let them do it. What is the worst that can happen? Now compare that against what you can gain from letting people have a bit more fun in your fights. I can all but guarantee that if you loosen up a bit fights will be more entertaining, paint a better picture in your head, and even probably run faster. In larger fights this may be harder to do, but that is also where the time saving aspect could become even more vital.
Step 2: Dish out the status effects
Most systems have various status effects that can be given to people. Knocked prone, disarmed, being thrown around. There are a good amount of them, they may require mechanical things being done to have them happen but by all means do them. Knock your players on their ass, then let them get back up and do it to them again. Their character might get frustrated, but if you start doing these things to them they are going to know that they can be done. This means they're a lot more likely to do them back. When everyone is knocking people down, disarming weapons, and this that and the other thing, you get a much more entertaining fight.
Step 3: Be descriptive
End of the round, after attacks, for monsters and players be sure to describe what happens. Describe the impact of the blows, describe the attacks, give visual references if you can. Stand up, act it out in slow motion. But be descriptive. If you paint the picture in their mind, they can follow a lot better. Remember when describing that less can be more, you don't want to go into a lecture for every attack, but you want to give them enough to be able to follow it in their head as more than just a string of mechanics and numbers. This makes it a bit more mentally taxing on you, but the rewards are more than worth it. Especially when you start to see the smiles showing up on players faces when you get into combat, as opposed to the "welp, here goes at least an hour of suck" that you get if you aren't being descriptive.
Step 4: Have FUN!!!!
Honestly, I can't stress this step enough, and it is the most important step. Have FUN with your combat. Send in an ogre, throw people around. Do things that may not be mechanically possible but are fun. A person favorite of mine with larger creatures is exactly as I said, grab player 1 throw him into player 2. Low damage, but 2 characters are tied up for a round as they have to unentangle themselves and stand back up. Meanwhile though, people get the idea that the monster is strong and dangerous while the 2-4 people still standing have to worry about that round where their friends can't help each other.
If you're having fun, it is going to come through and your players will pick up on it. Next thing you know, everyone is going to be having fun and trying to do all sorts of insane stuff to each other. What does that make? A fun, and potentially very cinematic combat that everyone can enjoy. Isn't that the point of what we're doing? Trying to have fun?
Step 4 really is the most important step, having fun with combat is more important than everything else I just listed here. You can have fun while doing the number crunching, while being descriptive, while giving status effects and they will all work out amazingly well. More importantly, if you're not having fun while doing these things, they simply put aren't going to work.
That being said, lightening up on rules, dishing out status effects unfairly, and having fun despite mechanics (or while ignoring them) may not be what your PCs want, or what they are comfortable with. Talk to them about this before you do it, tell them you want to try it and give it a shot for a session. If people are having more fun the old way where it is just mechanics, then that is what your group wants. However, if you do it right (and it may take a bit of practice to get right) you should be having much more entertaining fights in your game.