If I have an opponent's creature with a Mind Control attached to it, and then Cloudshift that creature, what happens? I know the enchantment will fall off, but Cloudshift specifically says 'return that creature to the battlefield under your control'. Does this mean I now control the creature without having to use the enchantment?
Yes. You will have permanent control of that creature unless your opponent uses another effect to take it back. The best move would be to wait until your opponent goes to kill the creature or destroy Mind Control and then Cloudshift in response.
You will have permanent control of the creature, the enchantment will be destroyed, and the creature will remain yours unless the opponent kills it.
If you're going to mind control to steal you should use act of treason instead and then cloudshift.
@justinthewhite2323, that would give you permanent control of a creature you only had for the turn.
Using Cloudshift on a creature that you have Mind Control led is only necessary as a defensive tactic. If they go to kill the creature or enchantment, you can cloudshift in response to save the creature/keep control of it.