First, the definition:
Whose is the possessive form of who.
"Of course I know whose fez it is," Amy said. "Why do you think I stole it?"
Who's is the contraction form of who; that is, a shortening of who is or who has.
"Who's this River Song, and how does she know so much about you?" Amy asked the Doctor.
"Who's got my fez?" the Doctor demanded. "Fezzes are cool."
The easiest way to figure out which to use is to say the sentence out loud. Can you substitute who is or who has? If so, use who's. If not, whose is your man.
See the original article, "When do you use 'who's' versus 'whose'?"
Also, see "Who's vs Whose" on Education Bug.