Nature of charge on conductive surfaces In our introductory physics class, we were being taught about charge density ($\sigma$). Our teacher told us that  a spherical conductor has a uniform charge density while conductors like a cuboidal conductor have a greater charge density at the vertices. In general, charge density is greater than average in uplifted areas. She also said that in conductors, charge was restricted only to the surface. I asked her why. She wasn't able to deliver a satisfactory answer. 
 A: Assume that there were charges in a conductor anywhere else than at the surface. Then they would repell eachother by electrostatic force until they couldn't move away further from each other, i.e. until they have reached the surface. In other words: charges move as long as they reach equilibrium. For a spherical conductor, equilibrium is realized for a uniform distribution on the surface, while for other shapes (like a cube) there will be regions of higher density. Another formulation of this equilibrium is the condition that the electric field vanishes inside a conductor. 
A: To develop on the "other formulation" mentioned by Frederic Brunner, which i consider to be a little bit more intuitive. First, we assume that the electrons of a metal are free enough so that any existing field will cause them to move on their own in direction opposite to the field.
Next, suppose we establish an external field that induces charge on the conductor or establish a field within the conductor by locally introducing the charge somewhere in the metal. From our first postulation, now that field exists within the conductor, the free electrons will start moving opposite to the field. They are free to move but not free to leave the surface. As they change their position, they also change the field within and outside the conductor by changing the charge distribution. This process will continue until there is no remaining field inside the conductor and therefore no 'incentive' for the electrons to move. At this point the entire conductor is an equipotential.
This also implies that there is no charge within the body of the conductor because if there were local charge densities, then a gaussian surface enclosing the local charged area would give a non-zero flux and therefore our postulation of no field And equipotentiality of the conductor is violated. Therefore the only charge distribution that can possibly make zero field inside is at the surfaces.
(If the establishment of zero field involves movement of charges, why does it always give a perfect arrangement? Nature doesn't calculate the best arrangement . Instead it allows the charges to move under the field and establish the correct solution by undergoing several possible states.)
This way of thinking is more advantageous as in certain conditions, the net charge maybe concentrated in a very small region (in certain external field conditions) where the intuition of the charge arranging on the surface because of repulsion breaks down. But in any case, (Electrostatic case), the field inside a conductor will always be zero and the charge distribution on the surface in specific way ensures that.
