An attractor is the state the dynamics tends to when ever it finds itself in the attractor's basin of attraction and further, the dynamics will remain in that state until it is pushed outside that basin. Take for example the Lorenz attractor: start it with some initial state. If that initial state is outside a basin of attraction, the dynamics may fly to infinity. Start the dynamics inside the basin of some attractor and the dynamics settles into a bound state, the attractor. Purturb it a little, hit it a little, it will react. If you don't hit it much, it would jump and fall back to the attractor. Hit it hard enough and push it outside the basin, then it will either fly to infinity or find another basin to settle into. Program the Lorenz attractor in Mathematica and work with it for some time. That's the best way to understand attractors.
This is my favorite reference:
"Perspectives of nonlinear dynamics" by E. Atlee Jackson
and the quintessential reference:
"Chaos and Fractals" by Peitegen
