The goal of the ADVENT/Alien project is the perfection of this entity.
Physically, it actually resembles your own Psionic soldiers, save for a shock of wild, white hair, and a detail-obscuring faceplate, and it's own unique Psionic Repeater rifle. In actuality, it is a hybrid creature: the genetic soup of countless human beings have been processed and integrated with the DNA of both the Ethereal elders and various alien races.
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. it does have a tendency of staying close with it's reactive teleports, giving you a chance to move in for a high-damage flank with other soldiers. It has the Dimension Rift ability, as well as Mind Control.
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 Captain, the Avatar will appear in response to this latest objective (and like the Codex 'brains', even if the mission ends with an evac, you do recover the corpse automatically). Fortunately, this will be the only 'random' encounter with it.
The only other time you'll encounter enemy Avatars 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.