what do you mean by band gap of a semiconductor?

Band gap is an energy gap where no electrons can reach,  in a semiconductor it is what separate the conduction band and the valence band, it is very wide in insulators ,absent in good conductors and very narrow in semiconductors.

 In semiconductors electrons do not move into the forbidden gap at absolute temperature but can only do so at a temperature below the melting point of the semiconductor.

 For intrinsic semiconductor the band gap determine the conductivity as electrons and holes can only be excited to cross over it. Though the thermal excitation aid the cross over.

If the band energy of a material is above 3ev , it is considered to be an insulator,  the conductivity of an extrinsic semiconductor is depend on the type of doping it has, though high temperature will facilitate fast electron mobility.