Enchanting an opponent works very much like enchanting a permanent. The Aura spell targets the opponent. When it resolves, it enters the battlefield "attached" to that player. Once it's on the battlefield, it no longer targets that player.

Psychic Possession's controller, not the enchanted player, skips his or her draw step.

If Psychic Possession's controller ever happens to be the player it's enchanting, Psychic Possession will be put into its owner's graveyard as a state-based action.

An Aura with enchant opponent can't be attached to a permanent.

In a multiplayer game, if the enchanted opponent leaves the game, Psychic Possession will be put into its owner's graveyard as a state-based action. If Psychic Possession's owner leaves the game, Psychic Possession immediately leaves the game; this is not a state-based action.