The Japanese word koan means "public notice." It is taken from the Chinese kung-an, which is a legal proceeding. Formally, koans are presented as cases, with a "main case" presenting a phrase or fragment of a story, a brief commentary, and a capping verse.
In the Rinzai school of Zen, a student is given a particular koan to "solve" in his zazen practice. Most koans involve a paradox that cannot be solved by reason or intellect. The resolution forces the student into a different level of consciousness or comprehension.
It's important to understand koans as a means for teachers and students to work together. In private interviews with the teacher called dokusan, the student asks for guidance or for approval of his understanding. The face-to-face work between student and teacher is an essential part of Zen training.