The Liar Paradox
A sentence that says "this sentence is false" is still true, but only if it's not true, which means it's actually false, unless you're reading this sentence, in which case it's true, but only for you.
Go here for more mind-bending paradoxes.
The Barber Paradox
A barber who says all the people in his town who are not shaved by him are not shaved by him. If the barber is not shaving himself, then he is one of the people not shaved by him, so he should shave himself, which means he is shaved by him, which is a contradiction.
Or, you know, just go here to ponder the unshaved.
The Sorites Paradox
What is the color of the sky when the sky is green? No, really, what is it?