In a conductor, classically, there is an inifinite amount of charges that can move throughout the material without any appreciable amount of interference. Any electric field applied throughout a conductor will create a force on these charges. This causes the charges to move, the negative charges move in one direction and the positive charges move in the opposite direction. The charges collect on the surfaces because that is the only barrier to their movements since they have free movement throughout the conductor. Since we now have a collection of opposite charges on the surfaces, these charges will create their own electric field in accordance with Coulomb's law. This field will oppose the applied field. As long as any net field exists in the conductor, charges will be moved because they will experience a net force. Once the charges have configured themselves in such a manner that the electric field produced by the charges perfectly cancels the applied field, then we now have equilibrium because there will no longer be any net fields or forces inside the conductor.
Since there are no net fields inside the conductor, there are no opposing forces for a single test charge when it is placed inside the conductor. Since there are no net forces, the amount of work it takes to move the test charge anywhere inside the conductor is zero. Thus, the electric potential difference between any two points inside the conductor is also zero (because electric potential difference is a measure of the amount of work required to move a test charge from point A to point B).
In actuality, a real conductor has a finite supply of electrons and the positive charges, the ions left behind when an electron is stripped from an atom, are immobile as they are held in the metal's lattice. However, the amount of electrons on the surface of a good conductor, like copper, is still astronomical. We can still only strip off the electrons from the surface of a piece of copper and move them to the opposite surface of the copper and be able to produce an electric field that is much higher than most fields we will ever encounter. So, the approximation of infinite charges to move about is a very good one since the limit of this approximation requires very very large fields. The free movement of the electrons is still a good approximation. The electrons will meet resistance, as they bump into the atoms in the lattice (Drude model) but in the static case this is immaterial. The electrons will dissipate very little energy, which will be supplied from the applied field, and they can still move to respond to the applied fields. With statics, these transients are ignored and indeed they occur very very quickly.
So, I guess the salient points on a physical description are:
1. Charges move about freely in a conductor and will move in response to an applied electric field.
2. The charges separate to create local areas of net charge which in turn create secondary electric fields that cancel out the applied fields.
3. The charges collect at the surface because that is the only boundary to their movement and because any net field inside the conductor will cause charges to move and create secondary fields fighting the applied field.
4. In the end, the net field inside the conductor is zero which means the potential difference between points in a conductor is also zero.
