(related to Definition: Paradox)
One famous example of paradoxes is the liar's paradox:
"This proposition is false".
If the proposition was true, then it would be false. But if it was false, then it would be false that it is false, thus it would be true.
Other famous example is the barber's paradox:
"I'm a barber, and I shave anyone who does not shave himself, and no-one else".
The question is, who shaves the barber? If he doesn't shave himself, then he shaves anyone, who does not shave himself, so he does shave himself. But if he does, then he is someone who shaves himself. But this cannot be since he only shaves persons who do not shave themselves.
Yet another example is the Pinocchio's paradox:
"My nose is growing."
Pinocchio's nose grows if (and only if) what he is saying is false. If his nose is really growing, then he must be lying. But then his nose cannot grow. It cannot grow either, if he is saying the truth.