Charged plasma pistol shot, followed by BOOM HEAD-SHOT!!
Another good way is to attack them right after they melee you or go behind for the assasination. You should get a beat-down if you just bash them in the back. They may be fast on the battlefield, but since they melee mainly in kicks, it kills their recovery time to get moving again. That is their only major weakness I have found so far. That is the best strategy for taking them out on Heroic or Legendary, since any normal attacks will kill them on Easy and Normal.
Different strategy - Well, in Halo reach the Plasma pistol and headshot may work but you have to have enough ammo or you could get kicked in the face before you can fire. I suggest using Plasma Grenades or a Fuel Rod cannon.
Quite simply, sustained fire from fully-automatic weapons will drop 'em pretty effectively, but attempting to close in on the Elites to spray them on high difficulties isn't recommended. This way is not as efficient as the above ways, but it is a good strategy for less-expierenced players on the easier difficulties, like Easy and Normal. The Heroic or Legendary, keeping your distance from the Elite and shooting it with a ranged weapon while your squadmates distract the enemy can be very effective. For multiple Elites on these difficulties, ranged power weapons and/or close-knit teamwork with your squad will allow you to defeat them, but make sure you have a lot of patience or just love to load the previous checkpoint constantly.