What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what is a non-integrable system.) In particular, is there a dichotomy between "integrable" and "chaotic"? (There is an interesting wikipedia article but I don't find it completely satisfying.)
Update (Dec 2010): Thanks for the many excellent answers. I came accross another quote from Nigel Hitchin is: "Integrability of a system of differential equations should manifest itself through some generally recognizable features:
the existence of many conserved quantities
the presence of algebraic geometry
the ability to give explicit solutions.
These guidelines whould be interpreted in a very broad sense."
(If there are some aspects mentioned by Hitchin not addressed by the current answers, additions are welcome...)