One reason that graphene has comparatively remarkable conduction properties is its band structure, and in particular the idea that single-particle states carry a pseudospin. This sounds like jargon, and until I'd heard Philip Kim talk about this, I hadn't fully appreciated how this works. The idea is as follows. One way to think about the graphene lattice is that it consists of two triangular lattices offset from each other by one carbon-carbon bond length. If we had just one of those lattices, you could describe the single-particle electronic states as Bloch waves - these look like plane waves multiplied by functions that are spatially periodic with reference to that particular lattice. Since we have two such lattices, one way to describe each electronic state is as a linear combination of Bloch states from lattice A and lattice B. (The spatial periodicity associated with lattice A (B) is described by a set of reciprocal lattice vectors that are labeled K (K'))
Here is where things get tricky. The particular linear combinations that are the real single-particle eigenstates can be written using the same Pauli matrices that are used to describe the spin angular momentum of spin-1/2 particles. In fact, if you pick a single-particle eigenstate with a crystal momentum \hbar k, the correct combination of Pauli matrices to use would be the same as if you were describing a spin-1/2 particle oriented along the same direction as k. This property of the electronic states is called pseudospin. It does not correspond to a real spin in the sense of a real intrinsic angular momentum. It is, however, a compact way of keeping track of the role of the two sublattices in determining the properties of particular electronic states.
The consequences of this pseudospin description are very interesting. For example, this is related to why back-scattering is disfavored in clean graphene. In pseudospin language, a scattering event that flips the momentum of a particle from +k to -k would have to flip the pseudospin, too, and that's not easy. In non-pseudospin language, that kind of scattering would have to change the phase relationship between the A and B sublattice Bloch state components of the single-particle state. From that way of phrasing it, it's more clear (at least to me) why this is not easy - it requires rather deep changes to the whole extended wavefunction that distinguish between the different sublattices, and in a clean sample at T = 0, that shouldn't happen.
A good overview of this stuff can be found here (pdf) in this article from Physics Today, as well as this review article. Finally, Michael Fuhrer at the University of Maryland has a nice powerpoint slide show (here) that discusses how to think about the pseudospin. He does a much more thorough and informative job than I do here.