a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory.

a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true.

At the most basic level, a paradox is a statement that is self contradictory because it often contains two statements that are both true, but in general, cannot both be true at the same time.

i.e. No one goes to that restaurant because it is

always crowded.

Paradox can prove to be very revealing about human nature and the way that we speak. If someone says to you "I'm a compulsive liar," do you believe them or not? That statement in itself is a paradox, because it is self contradictory, which is precisely what a paradox is.