Conductivity of a crystalline solid In a crystalline solid each atomic level 'splits' into n levels (n = number of atoms in the system). When the number of atoms is large each level becomes replaced by a band of closely spaced levels.
In a semi-conductor we have an empty "conduction band" and a fully occupied "valance band". Conductivity arises because electrons get excited to the conduction band.
Question: Why can't electron in the valence band freely move around and therefore conduct electricity? My question also applies to metals where the conduction band is already half-filled. What's special about this conduction band that allows electron to move around freely?
 A: Every energy level in a band has an associated momentum, and the total momentum of all the levels in a band is zero. Because the total momentum is zero there can be no net movement of electrons and hence no current. In effect, for every electron with momentum $p$ there is another electron with momentum $-p$ and they cancel each other out.
You can't change the momentum of any of the electrons in a filled band by applying an external field because all the energy levels are full. There are no empty levels for you to move your electron into. That means an external field cannot cause a net movement of electrons.
When you excite an electron into the conduction band it will go into a low momentum state, but there are available states above it with higher momenta. Apply an external field and you will move the electron up into a state with higher momentum that is lined up with the field. In this state the electron has a velocity aligned with the field so there is a net movement of electrons and therefore a current.
The same effect causes a current in metals. There are empty states for the electrons in the band to move into.
A: Because when a band is full, as in undoped semiconductors and insulators, there is no state available into which an electron can move (in $k$ space) to produce a conduction of electricity. This is due to the Pauli Principle applied to the electronic states of a crystalline solid.
If you where to supply enough energy to one electron in the valence band to overcome the energy gap separating it from the conduction band, then it can access available states and then promote electrical conduction. Thus, what is special about this latter band is that has available (empty) states where the electrons can go to.
