Why does NaCl (aq) conduct electricity but NaCl (s) does not? Why does sodium choloride in its solid state not conduct electricity, but in its aqueous solution state it conducts electricity easily?
 A: Usually in conductors we think of mobile charge carriers as being electrons, but in an aqueous solution they can be ions. The two different ions have different affinities for electrons and therefore end up with different, non-zero charges, which move in response to an electric field. In the crystal state, there are no free electrons, because the energy level structure of the crystal doesn't have any places for electrons to go.  In order for an election to move, it has to go into a previously unoccupied energy level, because of the Pauli principle.
http://www.bandstructure.jp/Table/BAND/NaCl.html
Have a look at the energy levels for electrons in the above link. The straight horizontal line indicates how far electrons can get filled up from the bottom. Above this line the probability of finding an electron with that particular energy is essentially zero. Now, because there are no available energy levels there (the wavy lines), there are no free electrons. The lower levels are all filled up, and so if you apply a little bit of energy those electrons, the is no where for them to go, and they stay put. If you apply a lot of energy to them, to where they can start to conduct, then you further ionize the solid and no longer have NaCl.
Also, the ions are fixed in place and can't move, so no conduction takes place until you melt it.
Tl;dr: there is a band gap at the Fermi level.
