Roughly speaking, emergence describes the phenomenon that systems often display novel behaviors that weren't previously observed in the system: the cooperation of its constituent elements produces a wholly new behavior.
Emergence (at least in some respects) opposes reductionism, which holds that all emergent phenomena can be explained by the forces acting on a systems constituent elements.
Emergence is often described in a sort of scientific chain: biology is an emergent property of chemistry, which is in turn an emergent property of particle physics. It's humorously illustrated in this Abstruse Goose cartoon:
Perhaps there's a bit of handwaving there at the end, but it captures the idea well enough.
One of the struggles of modern science is what to do about consciousness, a phenomenon that more or less appears to be immaterial. The search for material causes for thought have generally met with little success, though the study of quantum mechanics and emergence has provided a bit of hope for thorough-going materialists.
Consciousness might be an emergent property of the physical universe, while quantum physics may provide a convenient explanation for the mechanics of thought itself—one that classical mechanics has been unable to provide.
The basic problem with thought is that natural science suggests that for every event, there is a sufficient natural cause. f = ma, and that's that: thought doesn't enter into this equation. This may pose no immediate problem, at least until we consider that causality is something of an uninterrupted chain. Where does thought enter in?
I'm not sure that emergence can really answer that question, neither am I persuaded that quantum uncertainty effectively accounts for free will (unless you're willing to suppose that you and a rock have the same sort of free will).
On the other hand, it is consistent with experience that we effect material changes on the basis of thought, and one wonders perhaps if the physical universe isn't rather an emergent property of the mental.
