A koan is a teaching riddle used in the Zen Buddhist tradition, primarily that of the Rinzai school. Koans are unresolvable via conventional intellectual reasoning. "Sitting with" (meditating on) a koan, a student tries to break through rational, dualistic thinking to appreciate the true nature of the self and the universe.
The first koan many students encounter is known as Mu. A Zen master was posed the question, "Does a dog have Buddha-nature or not?", to which the master replied, "Mu" (not). This answer is a conundrum because the Buddha is reputed to have said that all beings have Buddha-nature.
Probably the most universally recognized example of a koan is: "What is the sound of one hand clapping?", commonly attributed to the Buddhist teacher Hakuin Ekaku. (Actually, the koan is "What is the sound of one hand?" "Clapping" may be implied.)