Possession is X.A.N.A.'s own ability to infect other systems like a virus. This has been developed so its spectres can meld with real objects, devices, and living things. In X.A.N.A.'s case, they allow it to have total control over the vessels down to their molecular structure, becoming an instrument of X.A.N.A.'s will.
X.A.N.A. is a multi-agent system that takes over other systems like a virus. By activating a tower on Lyoko, it uses its own digital impulses that it sends through networks to connect to machines and devices and possess them, becoming tools to control at will. When they emerge as spectres, they can do the same thing to real objects and living things when planted inside, turning them into unnatural threats to attack the Earth.
In A Great Day, it was revealed that going back in time doubles X.A.N.A.'s power. When it hijacked the program, each trip increased its power, to the point where it could possess humans, transforming them into pixelized beings that do its bidding on Earth.
In Mister Pück, it attempted to possess Odd and Ulrich, but failed. It then tried to possess Jeremie and succeeded. It was later learned that people who have been scanned into the Supercomputer and virtualized onto Lyoko at least more than once are immune to possession.
Anything that has been possessed by X.A.N.A.'s spectres can be modified in almost any way that it sees fit, gaining modifications that people consider to be "supernatural".
In Lyoko Minus One, it is revealed that the Scyphozoa, one of X.A.N.A.'s monsters, has the ability to implant X.A.N.A.'s virus inside of virtualized people with its power to manipulate digital memory, effectively possessing them. The Lyoko Warriors' immunity to possession in the real world doesn't apply in the virtual world, ensuring that this is the only guaranteed method to take hold of them.