Cyborg enemy number one
These are the guys you really fear of course; or at least the ones with plasma rifles. Fast, tough and aggressive, they'll attack you mercilessly and soon be laughing over your lifeless tin-plated body if you don't watch out. Luckily though, you can defeat them with greater intelligence and skill, combined with a wider range of weaponry. But it may still hurt.
Red, blue, gold, silver
The most common Elites have red or blue armour and carry plasma rifles or needlers. The reds are noticeably more tough and aggressive, and a red with a plasma rifle always gets my full attention. On Heroic he can cut through your shield in a few seconds, which means you'd better react fast to prevent it. If you're caught by surprise like this, you could try a 'charge and whack' if you're close to him. Otherwise, you'd best head for cover to consider the situation. As you go, either loose a few shots, or lob a grenade that will make him spend time jumping clear.
In levels featuring normal Elites, the split between reds and blues seems to be randomized but with a possible bias one way or the other. Legendary shows a strong bias towards reds (expect more reds than blues), Heroic shows no obvious bias either way, Normal shows some bias towards blues, and Easy has a strong bias towards blues.
Gold Elites with plasma rifles are even more fearsome than reds, and sometimes fire so aggressively that the weapon overheats, which is a nice opportunity for you to strike back of course. I'm thinking there of a fellow you meet in one of level 5's octagonal rooms, who tends to continue blasting away long after you've ducked your head back from looking through a window in the interior wall. When you hear the weapon overheat, pop out again and let him have it! On the other hand, gold Elites with plasma swords are something of a pushover. You should easily be able to kill them before they get in range for a good swing. A swing can be dodged if you're fast, but you don't really want to be trying it!
The other standard type of Elite you encounter is the cloaked silver Elite. Fortunately they don't have shields (apparently it's incompatible with the cloaking technology), so they go down quite quickly once you get a bead on them. Some of these guys walk around with glowing plasma swords, which somewhat defeats the cloaking idea!
Special ops Elites
In their shiny dark armour, the special ops Elites you meet late in the campaign can be tough work. Nastier than reds, they're a lot more aggressive with the needler, though as usual it's the plasma rifles you'll fear most. They also have an added dimension: they throw grenades!
Yes, despite the fact that all types of Elite carry grenades, only the special ops Elites seem to actually throw them. That seems odd and I can't help wondering why Bungie arranged things like that. Anyway, when the throws do come, they're low and fast with a waist-level hurling action, and unlike the Grunts, the Elites don't have the decency to shout out something loud and clear beforehand. There is a call, but it's brief and not terribly distinct in the heat of battle, so you really have to stay on your toes. The throw is sometimes strangely buggy; the grenade can seem to get left behind at his midriff for a moment, before whizzing away as if back on course.
Shooting at such an Elite tends to make him throw if he's not too close, which can be useful as a tactic to stop him shooting for a moment, to give you chance to strike back in some way (e.g. sending in a plasma ball to take his shield down).
Bye bye Mr Elite
The classic tactic of a plasma ball followed by a pistol headshot is particularly valuable against highly aggressive Elites with plasma rifles, as it minimizes your exposure. Taking him by surprise if possible, pop out from cover and instantly send off the plasma ball. Switching to your pistol, maybe pop back under cover momentarily while the shot is on its way. Once he's stunned, finish off quick while he's still dazed. All very neat when it goes to plan, though you could take some nasty fire if you fluff the headshot.
The needler is also a great Elite killer, though you tend to have to expose yourself longer to get off enough needles for the desired explosion.
Whack dodging and tempting
When fighting up close with an Elite, be ready for when he shapes up to whack you. Step back or to the side to dodge it whilst continuing your attack. You also have the opportunity to start administering a good whacking of your own. Once you become a confident dodger, you'll always be pleased to see an Elite shaping up to whack you, because you know it means curtains for him.
If you're up close to an Elite and in danger of getting cut to shreds by his plasma rifle, it can be worth moving in to deliberately tempt him into a whack, to stop him shooting. This buys you valuable time and should turn the tables nicely. I do quite a bit of this in tight situations; often while drilling him with an assault rifle. Actually, if you stay close, it's possible to tempt him into one whack after another, toying with him. The big buffoon just can't resist a good swipe!
Tag and pass
Sometimes you may need to deal a close Elite fast, due to other threats. This is where a 'tag and pass' move can be handy. Charge at him and tag him point-blank as you skim past, dodging the whack he'll probably try. Executed well, this is a smooth and classy move that'll make you feel good all over. Unlike the Elite, who'll be left with something nasty fizzing away on his person as he completes his swipe. Ha ha… yeah, that didn't go according to plan did it Mr Squid-Face? I use this move quite a bit in megabattle situations where I'm under heavy pressure from multiple Elites; especially in the far area defence for the rockslide megabattle.
Don't let him recharge
When an Elite's shield is damaged, he'll often take cover until it's recharged. You should be aware of when that's happening, and try to take advantage by either surprising him (such as when he's gone behind a big rock and you go around the other way) or moving somewhere where you can keep wearing him down.
Killing cloaked Elites
Killing a cloaked Elite who has a plasma rifle can be tricky, especially on the higher difficulty levels where it's dangerous to simply blunder forward for a close-range blasting match. A grenade can be a fine way of giving him a nasty shock if you've got an idea of where he is. But in general the safest thing to do involves staying well back, patiently watching for an outline. Possibly loose off an occasion shot if you think it might tempt him into your line of sight. Once you see the outline (relax your eyes, don't strain), the high shot speed of a pistol is ideal for getting a hit before he moves again, and you might even manage a headshot. If you don't have a pistol, a plasma rifle is a fair second choice. Once he takes damage and starts to become visible, try to keep up the pressure to keep him showing, perhaps even moving in for the kill.
In a few situations you have the opportunity for being far enough away that you can use a sniper rifle instead of a pistol. I'm particularly thinking of a bridge in level 5, where you can retreat right back to the far end and watch for advancing outlines through the sniper scope.
Another way of tackling cloaked Elites is to spray plasma about the place hoping for a hit, to cause momentary visibility. As soon as you see one, focus your fire on him, ideally switching to a pistol.
Of course, if you have a cloaked Elite calmly pacing about a room unaware of your presence, things are a lot easier. You might creep up and whack him on the head or tag him with a plasma grenade; or calmly line up a pistol headshot.