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 also has the Rift ability, as well as Mind Control and Psi Lance.
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.
After killing it, you are prompted to complete any and all other Blacksite/Avatar-related Shadow Project research, then doing a final project to unlock the last two missions: one to hijack the main ADVENT control tower to expose the Administration's evil (you will have only 3 soldiers on this mission), and another to assault the underwater alien base, in perhaps a callback to the battle of T'leth. In this final mission, your mind will be 'riding shotgun' in the Avatar corpse, and will be a seventh soldier in this final battle: much like the Volunteer, Your Avatar must live to the end of the final battle.
At the end of the Alien Base, you will encounter a room with many areas of cover, and more portal gates. The first one will be on a raised platform up high, and will move to engage you when spotted. Then as the turns progress, more will spawn in from the other portals, accompanied by alien pods of varying composition and threat level. Killing all the Avatars wins the game, so keeping away from the other aliens to focus on the Avatar may work as a last stand.