Consider this diagram:
This is a benzene molecule, which is basically the smallest unit of graphite that you can have. It is what is called an aromatic ring. As you can see, there are six carbons in a coplanar hexagonal arrangement. The left figure shows the Lewis structure of the ring. Notice how half of the carbons are double-bonded to each-other and half are single-bonded.
The right figure shows the actual arrangement of the orbitals in the double bonds. The orbitals that form the second bond are called pi orbitals and each contains one electron. In reality, the electrons can travel freely around the ring. The double bonds shown in the left figure don't actually exist. What actually happens is that there is one-and-a-half bonds between each carbon and its neighbor, and the electrons are delocalized, traveling freely from one pi orbital to the next.
What this means is that in the following figure the two structures are one and the same; they are not rotations of each-other. If you were to replace two adjacent hydrogen atoms on the ring with tritium, there would be simultaneously a single and double bond between those two carbons. When this happens, the two different figures are called "resonance forms" and they are both theoretical concepts. The reality is that there are 1.5 bonds between each carbon.
And so this is the result:
, but I'm talking about benzene. What does this have to do with graphite?
Well, graphite is a bunch of benzene rings that are all linked together, like this:
Never mind the vertical bonds. They represent weak interactions between adjacent sheets of graphite. But this illustrates how a single electron could quickly travel from end of the sheet to the other. But only in the plane of the sheet. Current cannot cross through the sheet because it would then have to pass through a region of zero electron density and that is not allowed.
Yay for college organic chemistry!