I think sgstudent is asking how the electrons move inside a conductor, not asking about conventional current flow vs electron flow, or energy flow.
Conductors, like everything else, are made of atoms. atoms have a positively charged nucleus surrounded by negatively charged electrons. In a conductor the outer shell of electrons surrounding the nucleus can pass from one atom to another. These electrons are called valence electrons, the more valence electrons a certain material has, the better it is at conducting electricity. Remember that, a conductor is already full of electrons, the battery isn't spitting electrons out into it and then sucking them back up. The battery "pumps" the electrons in the conductor around the circuit.
These electrons do move very slowly, this speed is called drift velocity. The energy however moves much quicker. There are lots of analogy's to explain this, a classic one is marbles in a tube: imagine marbles lined up next to each other in a tube, push the first one and the last one seems to instantly move. The marbles may be moving slow but the energy has traveled fast. (not instantly though. It may appear that the last marble moved instantly but the energy actually traveled at the speed of sound for this system).
I'm not sure how valid this analogy actually is though. I don't think the energy through a circuit travels at the speed of sound. Here's how I think of it:
When an electron moves, the electric field it creates moves, causing there to be a net force on the otherwise equilibrium system of electrons. This causes these electrons to move and therefore their electric fields move. The electric field redistribution can travel very quickly around a circuit.
The electric field at some point in the circuit, multiplied by the charge at that point in the circuit, multiplied by the distance moved by that charge due to the electric field, equals the work done by the electric field on that charge. You can see if no charges are moving, no work is being done. You can also see that the electric field redistribution causing the charges to be moved could cause work to be done far away from the original electric field redistribution very quickly. This is why energy flows around a circuit faster than the electrons do.
But to simply answer your question "how does current flow?": charges (electrons) inside conductors are pumped around the circuit because of a potential difference. the electrons can and do pass through the battery(in the form of anions).