Not so orderly as grooves on a record. Have you ever scribbled with a pen for a long time, making loop upon loop that all cross over a central point? Some loops are huge, others small, all eventually make their way back to the dark inky blotch in the middle. Something like that.
Sadly I don't remember much about them, but the Wikipedia article on attractors is actually pretty nice. Seems like attractors are just some set of numbers for which a system tends toward. Maybe the fixed-point attractor is a better descriptor of your original idea: the system tends toward one fixed point and oscillates around it chaotically. I first read about attractors in high school in the book Chaos by James Gleick, which is a fantastic book I highly recommend for this sort of stuff.