The koan is not a conundrum to be solved by nimble wit. It is not a verbal psychiatric device for shocking the disintegrated ego of the student into some kind of stability. Nor, in my opinion, is it ever a paradoxical statement except to those who view it from outside.
--Ruth Fuller Sasaki

Victor Sogen Hori, who like Ruth Sasaki spent years in traditional koan study, cites Dogen.

Dogen attributes to the "scatterbrained" the same two views of koan and _kensho_--that "incomprehensible utterances" are merely skillful means to cut off the tangling vines of discriminating thought in order to bring one to the great enlightenment, and that the great enlightenment itself is noncognitive, something "prior to the emergence of any incipient sign." Dogen heaps scorn on this view.

Just in case we don't get the point, Victor Hori also writes

But a pure consciousness without concepts, if there could be such a thing, would be a booming, buzzing confusion, a sensory field of flashes of light, unidentifiable sounds, ambiguous shapes, color patches without significance. This is not the consciousness of the enlightened Zen master.

As with any question, there's a short answer, a long answer, and the truth. The short answer is that a koan is something that a monk works with very intently until she realizes that the koan is herself working with the koan. To put it differently, she becomes one with the koan. For the long answer, read The Koan: Texts and Contexts in Zen Buddhism, a collection of scholarly essays edited by Steven Heine and Dale S. Wright, or Ruth Sazaki's classic book The Zen Koan. But if you want to know the truth, it is this: Cheese is made from milk.

The Shambhala Guide to Buddhism and Zen defines a koan as "a paradox, i.e. that which is 'beyond' (Gk., para) 'thinking' (Gk. dokein), which transcends the logical or conceptual. Thus, since it cannot be solved by reason, a koan is not a riddle. Solving a koan requires a leap to another level of comprehension. . . . Since the koan eludes solution by means of discursive understanding, it makes clear to the student the limitations of thought and eventually forces him to transcend it in an intuitive leap."

A koan is a tool for zen practice. It is a question that forces us to face the lack of a rational answer. Koan practice is an especially prominent practice in the Rinzai tradition of Zen Buddhism but is also practiced in other Zen schools.

For futher reading as an exploration of koans and their role in Zen practice and tradition, consider checking out The Blue Cliff Record, The True Dharma Eye, or Bring Me the Rhinoceros.

With peace,

-Sapito