TITLE: What is an integrable system?
QUESTION [149 upvotes]: What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) 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 across another quote from Nigel Hitchin:
"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 would be interpreted in a very broad sense."
(If there are some aspects mentioned by Hitchin not addressed by the current answers, additions are welcome...)
Closely related questions:

What does it mean for a differential equation "to be integrable"?
basic questions on quantum integrable systems

REPLY [2 votes]: "All integrable Hamiltonian systems are alike, while each nonintegrable one is nonintegrable in its own way", Valerij V. Kozlov (after L. N. Tolstoi of course), Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996