Answer:
There are many kinds and definitions of paradoxes, and many are variations of others. The most famous is Russel's Paradox, which revolutionised logic and set theory.
Russel's: Does the set of sets that don't contain themselves contain itself?
Barber: The barber shaves men who do not shave themselves. Who shaves him?
Liar's: "My fellow citizens, I am now lying to you"
Coastline: The smaller the ruler, the longer an island's coast is.
Arrow information: You need to understand the info you're buying before you buy it.
...and many more. There are economic, political, topological, and even astronomical paradoxes that are actually quite entertaining.