I love combat, so I have quite a few thoughts here

1) The stakes: Before combat even starts, make sure the game as a whole is immersive and interesting. Players have to really care about their characters and the game world in general so that they care about the outcome of the combat. If players aren't fully invested in the world then combat will be boring just like everything else is.

2) Danger: The players have to feel that there's a real possibility that they will lose the combat, or that some other bad thing will happen. If they are virtually guaranteed to win a combat, it quickly becomes a matter of going through the motions. Of course, not every combat has to be something that could be a TPK (if it is, then you'll kill players often which you might not necessarily want to do) but even if the enemies haven't got much chance of killing the players, there should be some other kind of pressure on them, like wasting their time, their resources, or perhaps the players want to take the enemies alive for some reason. By the same token, combat is only exciting as long as there's a chance for something bad to happen to the players; which means that as soon as the players have clearly turned the tides in the battle and the enemies no longer have much chance to do anything to the players, get the battle over with as quickly as possible. Have the enemies surrender or run away once it becomes clear they can no longer win. Mop up is boring so get it over with asap.

3) Depth: Combat, like all aspects of gaming, is interesting only insofar as it presents a series of interesting choices. Every round of combat should offer the player difficult choices of real consequence. This means that your system needs to have a certain amount of depth; players need to have options available to them that are equally useful in different circumstances. Players in general should feel that their choices matter; skill in combat should matter. Not everyone agrees with this, but if you want to make combat interesting it does have to be the case that players can be 'good' at it, and that has to mean that players can be 'bad' at it too. Players should feel over time that they are getting 'better' at combat, just as their characters in game accumulate experience and get better too.

4) Pace: combat in real life is chaotic and fast paced; you want to give that feeling to players so make sure you keep the pace going in your combats as well. As a rule, I give players only 10 seconds to declare an action on their turn and sometimes I feel like it should be 5 seconds. I keep to the same rule myself with bad guys. Sometimes this means players or enemies make mistakes and do something sub-optimal. Good! People don't always behave perfectly rationally and strategically in combat and the ability to think quickly on your feet should be rewarded anyway. Nothing takes people out of the excitement and chaos of combat faster than some guy agonizing for a minute or more over which spell to cast or whether to move here or there or whatever.

5) A Twist: every combat should have some unique, unexpected factor to it. A 'twist' that makes it memorable. A lot of times, good players will provide a twist on their own, but as DM you can't just sit back and wait for players to make it interesting; you need to be pro-active here. Some examples of twists: enemy reinforcements unexpectedly arrive; an enemy transforms into a more powerful form; a unique setting or setting change; a friendly NPC turns heel and backstabs the party; a secretly cursed item suddenly activates its negative effects; NPC hostages are threatened by the bad guys; a neutral third party appears and interferes; a time limit--a bomb is about to go off, etc; a unique mechanic--aerial combat, a car chase, etc.

6) Evolution over time: Combats should evolve and change over the course of the game as players and enemies acquire more and more powerful abilities. At level 1, when even a small group of goblins can present a deadly threat, combat is fairly straightforward and much of the strategy revolves simply around getting the drop on them. At the end-game stage when players are going after demi-gods combat has a much different feel; the strategy now revolves also around careful and judicious use of limited-use abilities; movement is not just running around but also tons of instant movement type abilities; buffs and debuffs are super important, AOE and DOT abilities are common and important, the battlefield can be changed with different kinds of walls or AOE ongoing effects, and so on.