Answer:
A koan is a question or statement usually associated with Zen Buddhist practice posed to lead a student to comprehend without thinking. A mantra is a chant, prayer or Vedic hymn (all terms approximate) which is repeated multiple times to facilitate a state of mind usually as part of meditative practices. Mantra should not be confused with mandala a symbolic geometric drawing. A mantra is in some ways a verbal mandala.
The term "koan" is from the Chinese kung-an, which literally means "public cases." Koans in zen training are enigmatic expressions that have come down from centuries of dialogues between masters and their students.
Koans are thought of in two ways: as verbal tools to bring about spiritual awakening and as expressions of the awakened minds of masters.
Koans are verbal signposts or pointers. Taken in isolation, they make no sense. The conceptual contradictions are useful in assisting trainees is going beyond mere conceptual thinking to awareness of the totality of what-is.
There are different kinds of koans. A good introduction is THE KOAN edited by S. Heine and D. S. Wright.