Related question on EE: Does perfect insulation exist? (especially the part about vacuum)
Insulators and conductors
The property of a material to carry charges from one point to another is what electric current is.
The difference between insulators and conductors lies in the electron band structure they posses. In conductors the Fermi-level (thermodynamically probable energy of an electron) is in a conducting(valence) band. This means most electrons will be ready to move and electricity will flow easily.
In an insulator the Fermi level is far away from the next conduction band, this means that very few electrons will be ready to move(because of a low probability being at a specific state). Other electrons can only be excited by very strong fields or other circumstances.
Therefore the conduction in an insulator can be described as by chance. As air is an insulator this is the case.
How does pressure change this behavior?
Very low, and very high pressure will reduce the number of charged particles that get transported. The first case is because there are only few particles, the second is because it is too stuffed and they collide and loose energy all the time. Actually you shouldn't look at pressure, but the product of pressure and electrode distance, as their product is the thing that is important. That means low and high pressure reduce conductivity.
Except for probabilities, in real life you have cosmic and other rays that can ionize air molecules. This is actually a very important part of the conduction mechanism for gasses, and can't be neglected.
Breakdown laws (but for voltages lower than the breakdown voltage)
As you were only interested in conductivity I will just tell you a few keywords regarding the other aspects of conduction during breakdown: Pashen's law, Townsend Mechanism, Streamers.
These mechanisms can be fed parameters which won't result in breakdown. In these cases they will describe the conduction mechanism (to some degree).