The Elders' own bodies have degenerated to a substantial degree. The goal of the ADVENT/Alien Avatar project is the perfection of this entity - a new perfect body for an Ethereal to inhabit, with high psionic ability, speed and agility, and a blank slate with no existing mind to interfere with the Ethereal's control. Not only would this allow the Elders to escape their atrophying bodies, it would also amplify their psionic powers to a degree that would make them unstoppable. While the Avatars fought in-game are mere prototypes, they are no less dangerous for it.
Physically, it actually resembles your own Psionic soldiers, save for a shock of wild, gravity-defying white hair, a detail-obscuring faceplate, and its own unique Psionic Repeater rifle. In actuality, it is a hybrid creature: the result of rendering down countless human beings, especially those with psionic potential, into genetic soup, then adding alien and Elder DNA. The ghostly form of the Ethereal inhabiting the Avatar's body can be seen behind it.
Like the Codex, it is capable of teleportation, and will do so after every time it takes damage. Fortunately, it is not capable of cloning itself. It makes up for this by having a much heftier health pool and innate health regeneration. it does have a tendency of staying close with it's reactive teleports, sometimes giving you a chance to move in for a high-damage flank with other soldiers. It has the Dimensional Rift ability and Null Lance to inflict direct AoE damage, as well as Mind Control. Avatars are immune to Disorient, so their Mind Control is hard to break - you may want to bring a Solace Psi Operative along when you're expecting to face one.
Your first encounter with this being will be after you're prompted to Skulljack a living Codex: much like how you encountered the first Codex after Skulljacking an ADVENT Officer, the Avatar will appear in response to this latest objective (and like the Codex 'brain', even if the mission ends with an evac, you recover the corpse automatically). Fortunately, this will be the only 'random' encounter with it.
The only other time you'll encounter enemy Avatars in the campaign is in the final room of the final mission. Three alien Avatars will attack you, and defeating them completes the game. In the final mission, you also control an Avatar of your own, inhabited by you, the Commander. This Avatar must survive to the end of the mission.
Multiple Avatars will also appear in the final missions of each of the Tactical Legacy Pack's mission sets.