F.T.W. means Fuck The World. Its often used by the lower class of youth. Its used by people pist off about society, pist about the way things are today. Angry about whatever it may be. F.T.W. is simply stating how a person may feel about the world.
kid on the street: F.T.W. its all shit to me!
kid's friend: yeah the world is so fucked up! F.T.W.!