Band theory is the theory that explain the electronic conduction in metals and the electron-hole conduction in semiconductors. Basically it applies to all solids, and the formation of bands can be described by the following phenomena. When 2 atoms (metallic or semiconductor) form a solid, they share together electrons and form covalent bonds. This means that the valence electrons of both atoms become to belong to both atoms equally. The form of the atomic orbitals (which covered only the space of an atom) is modified into a molecular orbital which covers now the space of both atoms. Otherwise described, from two identical energy levels (each belonging to one atom) we obtain two separated but very close energy levels (belonging to both atoms).
Now if a large number of atoms come close together to form the solid, the same things as for 2 atoms happens. First the valence electrons from each atom, become to belong to all atoms equally. A molecular orbital which extends over all atoms is formed. All the identical energy levels in individual atoms, splits into separated but very close energy levels. This is what a band is named. If you want to say it differently, bands are for solids what energy levels are for individual atoms. There are allowed bands and forbidden bands.
A band can be only partial filled with the valence electrons from individual atoms. This is the case of metals. Only a very small energy is needed for an electron in a lower energy state to jump above in a free state and thus conduct.
Or a band can be completely filled with electrons. This is the case of semiconductors and insulators. If the energy gap to the next band is relatively small, an electron can jump into the next allowed band and conduct. This is the case of insulators. If the energy gap to the next allowed band is very high, the electrons in the lower band can not jump into the next allowed band and thus electric conduction can not take place. This is the case of insulators.