A paradox is a statement that seems absurd or self-contradictory but may turn out to be true, such as saying a restaurant is crowded so no one goes there. At its core, a paradox contains two statements that are both true but cannot be true at the same time. One example is someone claiming to be a compulsive liar, which is paradoxical because it contradicts itself.

1.  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.

2.  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.

3.  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.