What does "optical conductivity" mean? Does it just mean "AC electric conductivity"? If so, why have a special name for it, and why mention optical specifically?
The wikipedia page on it is very sparse. This (warning, PDF) document just says:

The term “optical conductivity” means the electrical conductivity in
  the presence of an alternating electric field.

Which sounds exactly like AC conductivity to me.
Is it different? If not, then why the special name?
 A: Optical conductivity and AC electric conductivity experiments are indeed quite similar. However, they operate in different frequency regimes and measure slightly different quantities. 
Optical conductivity refers to an experiment using light, such as a reflectivity measurement and then using a Kramers-Kronig transform to deduce the real part of the transverse conductivity, $\sigma^T$. The transverse conductivity is measured because the direction of propagation of the photon is perpendicular to the electric field. Another way to measure the transverse optical conductivity is through a light transmission experiment. In both cases, the conductivity is determined over the THz frequency range.
On the other hand, the AC electrical conductivity usually involves some sort of electrical circuit and usually measures the real part of the longitudinal conductivity, $\sigma^L$, in the MHz frequency range. In this case the probe and the electric field point in the same direction.
All in all, the measurements obtain slightly different experimental quantities and also over different frequency ranges. I say slightly different here because in the limit that the wavevector $q\rightarrow0$, the regime that we are in for the above experiments, it is expected that the longitudinal and transverse response become indistinguishable (the only known exception to this rule is in a superconductor). 
