To rebut is to argue against something. If your parents say you're too young and irresponsible to drive, you can rebut their claim by ticking off examples of your responsibility.
When you argue against something, you rebut that position or argument. Your school's principal might rebut your teacher's argument that the class is overcrowded by pointing out that there could legally be five more kids in the class. The teacher could rebut the principal's rebuttal by observing that there aren't enough books or seats for the kids in the classroom now. Rebut comes from an old French word rebuter, meaning "to thrust back."