Spells are under your control the moment you cast them. If that spell becomes a permanent (such as a creature) you keep control of that permanent (unless an effect says so) until it leaves the battlefield
When a player is said to control a permanent, said permanent must have it's choices made by the controlling player. The controlling player chooses when/if to use it's activated abilities, (if it's a creature) when/if it will attack, and when/if to sacrifice the permanent among other things. The controlling player doesn't necessarily have to own the card.
Note that some spells or effects, such as Willbreaker's ability, let you gain control of cards of your opponent.