In solids, the valence band is the highest range of electron energies where electrons are normally present at absolute zero. In semiconductors and insulators, there is a band gap above the valence band, followed by a conduction band above that. In metals, the conduction band has no energy gap separating it from the valence band (basically, this is correct only for semimetals. All solids have forbidden energy levels between the energy bands). The rest of this article refers to the valence band in semiconductors and insulators.
Semiconductors and insulators owe their low conductivity to the properties of the valence band in those materials. It just so happens that the number of electrons is precisely equal to the number of states available up to the top of the valence band. There are no available states in the band gap. This means that when an electric field is applied, the electrons cannot increase their energy (i.e., accelerate) because there are no states available to the electrons where they would be moving faster than they are already going.
There is some conductivity in insulators, however. This is due to thermal excitation—some of the electrons get enough energy to jump the band gap in one go. Once they are in the conduction band, they can conduct electricity, as can the hole they left behind in the valence band. The hole is an empty state that allows electrons in the valence band some degree of freedom.
It is a common misconception to refer to electrons in insulators as "bound"—as if they were somehow attached to the nucleus and could not move. Electrons in insulators are free to move. They are also delocalized, having no well-defined position within the sample.