Physicalism can most accurately be defined as the metaphysical thesis that everything is either physical or "supervenes" on the physical (Stanford Encyclopedia of Philosophy). In simpler terms, one can say that it is the idea that everything that exists can be described either as a physical thing (i.e. matter or energy) or a consequence of the interaction of physical things. I happen to be a tentative physicalist. It's my working assumption, although I don't claim to be even remotely certain, and I'm very doubtful that we could ever come to know that something non-physical exists or not. I guess one could say that I am a physicalist in the sense that matter and energy (and the things that supervene upon them) are the only things I believe to exist, as opposed to believing that matter and energy are the only things that exist.
Explaining consciousness is perhaps one of the greatest challenges to a physicalist at this point, which is one of the major contentions that opponents of physicalism have against the principle. Being a tentative physicalist, I really only feel the need to demonstrate that it is possible for physicalism to account for consciousness as opposed to verifying the thesis. That, along with the application of Occam's Razor, is enough to maintain my skepticism of claims to anything not supervenient to the physical. I fully admit that neurology is much, much too young a field to really help verify that physicalism does account for the mind, and thus will refrain from any arguments about how brain functions correlate quite well to mental states. I assume that both physicalists and its opponents accept this as a matter of fact, so I don't believe it to be of any use to argue on this point.
So, there are really only two questions to be answered here. First, how do physicalists suppose that mental states arise from physical phenomena? and second, is it possible? Here, I'll only be answering the first (for now).
Physicalists most often suppose that the mind is a product of complex interactions happening within the neural network of the brain. The neural network of the brain, of course, is purely physical. Neurons are made of atoms, and neural networks are essentially large collections of neurons that interact with each other through chemo-electrical connections. This, according to the physicalist, is enough to account for the phenomenon that we call the mind. One may assert that the mind, or certain aspects of the mind, have no physical properties and thus clearly cannot be physical, but this, in my opinion, doesn't take into account the idea of supervenience.
Physicalists rarely assert that the mind is a physical object, but instead assert that it supervenes on the physical. This idea of supervenience seems a little complicated, but it really is quite simple. To say that a set of properties (A) supervenes on another set (B), it simply means that a difference in A-properties requires there to be a difference in B-properties, or that a similarity in B-properties require similarities in A-properties (Stanford Encyclopedia of Philosophy). It's important to stress that for something to supervene on another thing, they must not only be correlated but must also be causally linked (a change in B must cause a change in A). So, the physicalist believes that if two minds are different in a particular way, the two brains must be different in a particular way; and if two minds are similar in a particular manner, then the two brains must be similar in a particular manner; and that the similarities and differences of different minds are solely caused by similarities and differences within the brain.
The mind can then be described as a complex pattern that arises from interactions within the neural network of the brain. The staggering complexity of the brain is thought to create processes that don't seem to be able to be pinned down to any particular simple physical interaction, not much unlike how a school of fish seems to behave in a fashion that goes well beyond the relatively simple behaviors of any particular fish. And, just as it seems as though the behavior of fish in large groups tend to cause the movements of the school as a whole, it is thought that the operations of the physical components of the brain tend to cause mental operations to occur. These mental operations aren't "physical" in the most concrete sense of the word (they aren't physical objects), but they are not non-physical either. They instead is supervenient or consequent of the physical, or a process of the physical.