Suppose there is a town with just one barber, who is male. In this town, every man keeps himself clean-shaven, and he does so by doing exactly one of two things:
shaving himself; or
going to the barber.
Another way to state this is that “The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves.”
From this, asking the question “Who shaves the barber?” results in a paradox because according to the statement above, he can either shave himself, or go to the barber (which happens to be himself). However, neither of these possibilities is valid: they both result in the barber shaving himself, but he cannot do this because he only shaves those men “who do not shave themselves”.