This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
Chaos is apparently unpredictable behavior arising in a
deterministic system because of great sensitivity to initial
conditions. Chaos arises in a dynamical system if two arbitrarily
close starting points diverge exponentially, so that their future
behavior is eventually unpredictable.
Weather is considered chaotic since arbitrarily small variations in
initial conditions can result in radically different weather later.
This may limit the possibilities of long-term weather forecasting.
(The canonical example is the possibility of a butterfly's sneeze
affecting the weather enough to cause a hurricane weeks later.)
Devaney defines a function as chaotic if it has sensitive dependence
on initial conditions, it is topologically transitive, and periodic
points are dense. In other words, it is unpredictable, indecomposable,
and yet contains regularity.
Allgood and Yorke define chaos as a trajectory that is exponentially
unstable and neither periodic or asymptotically periodic. That is, it
oscillates irregularly without settling down.
sci.fractals may not be the best place for chaos/non-linear dynamics
questions, sci.nonlinear newsgroup should be much better.