A koan is a puzzling or illogical question, statement, or story to help the Zen student transcend logic, intellect, and notions of duality.
Koans seem silly or absurd precisely because they are undermining preconceived notions. To take the absurd seriously is to foil logic.
Knowing the intellectual answer to a koan, or someone else's answer, is not knowing the answer to a koan. Each individual must perceive the answer on an internal and intuitive level.
Traditionally, a Zen teacher assigns an appropriate koan to a student who periodically meets in private with the teacher to offer answers. The qualified Zen teacher can tell whether the student has truly understood the koan or not.
Some of the more famous koans are "What is the sound of one hand clapping?"; "Does a dog have Buddha nature?"; "Why did Bodhidharma come from the West?"; and "What is Buddha?"