a strange attractor
An ordinary attractor is a tidy resting place: a single point where motion stops, or a closed loop the system circles forever. A strange attractor is the bizarre third option that chaos lives on. The system is still drawn toward it and trapped on it forever, so it really is an attractor — but the motion on it never settles to a point and never closes into a loop. Instead the trajectory wanders endlessly over an intricate, infinitely-detailed shape, never repeating, never escaping. That object is a strange attractor.
What makes it 'strange' is twofold. Geometrically, it is a fractal: zoom in and you keep finding the same layered, filamentary structure at finer and finer scales, so it has a fractional dimension — more than a curve, less than a filled solid. This fractal layering is the visible record of a single mechanism, 'stretch and fold': the flow stretches a blob of nearby states apart (that is the sensitive dependence, the positive Lyapunov exponent) but, because everything must stay bounded, it repeatedly folds the stretched sheet back onto itself, like kneading dough. Endless stretching-and-folding manufactures the infinite fractal detail and, simultaneously, the never-repeating chaotic motion. Dynamically, then, a strange attractor carries chaos: at least one positive Lyapunov exponent, bounded but aperiodic trajectories, sensitive dependence.
Strange attractors are the geometric home of chaos — the answer to 'where does a chaotic trajectory actually live?' The Lorenz attractor, with its famous two-lobed butterfly shape, was the first one discovered, and the Henon and Rossler attractors are other classic examples. They matter because they show that long-term behaviour can be permanent and structured yet utterly unpredictable in detail: the system reliably ends up on the attractor (you can predict the shape), but you cannot say where on it the state will be far in the future (you cannot predict the path).
The Lorenz attractor: integrate the three Lorenz equations at their classic parameters and plot the trajectory in space. It traces out two spiral wings joined at the middle, looping a few times around one wing, then crossing unpredictably to the other, forever — a bounded butterfly-shaped fractal on which the motion never quite repeats.
The Lorenz butterfly: a bounded fractal set on which trajectories loop and switch wings forever without repeating.
Not every attractor with a fractal shape is dynamically chaotic, and the two words 'strange' (fractal geometry) and 'chaotic' (positive Lyapunov exponent) are, strictly, separate ideas — rare 'strange nonchaotic' attractors exist. In ordinary usage, though, strange attractor and chaotic attractor are taken to mean the same thing.