This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
Informally an attractor is simply a state into which a system settles (thus
dissipation is needed). Thus in the long term, a dissipative dynamical system
may settle into an attractor.
Interestingly enough, there is still some controversy in the mathematics
community as to an appropriate definition of this term. Most people adopt the
Attractor: A set in the phase space that has a neighborhood in which every
point stays nearby and approaches the attractor as time goes to infinity.
Thus imagine a ball rolling inside of a bowl. If we start the ball at a point
in the bowl with a velocity too small to reach the edge of the bowl, then
eventually the ball will settle down to the bottom of the bowl with zero
velocity: thus this equilibrium point is an attractor. The neighborhood of
points that eventually approach the attractor is the basin of attraction for
the attractor. In our example the basin is the set of all configurations
corresponding to the ball in the bowl, and for each such point all small
enough velocities (it is a set in the four dimensional phase space [2.4]).
Attractors can be simple, as the previous example. Another example of an
attractor is a limit cycle, which is a periodic orbit that is attracting
(limit cycles can also be repelling). More surprisingly, attractors can be
chaotic (see [2.9]) and/or strange (see [2.12]).
The boundary of a basin of attraction is often a very interesting object
since it distinguishes between different types of motion. Typically a basin
boundary is a saddle orbit, or such an orbit and its stable manifold. A crisis
is the change in an attractor when its basin boundary is destroyed.
An alternative definition of attractor is sometimes used because there
are systems that have sets that attract most, but not all, initial conditions
in their neighborhood (such phenomena is sometimes called riddling of the
basin). Thus, Milnor defines an attractor as a set for which a positive
measure (probability, if you like) of initial conditions in a neighborhood are
asymptotic to the set.