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In solid-state physics, an energy band is a continuous range of values of energy that an electron may or may not have. Bands may be either "allowed bands" (if electrons may occupy them) or "forbidden bands" (if they may not).
As isolated atoms are brought closer together to form a solid, various interactions occur between neighbouring atoms. The forces of attraction and repulsion between atoms will find a balance at the proper interatomic spacing for the crystal. In the process, important changes occur in the electron energy level configurations and these changes result in the varied electrical properties of solids.
Qualitatively, we can see that as atoms are brought closer, the application of Pauli's Exclusion principle comes into picture. Thus, there must be a splitting of the discrete energy levels of the isolated atoms into new energy levels belonging to the pair rather than to individual atoms.
In a solid, many atoms are brought together, so that the split energy levels form essentially continuous bands of energies.
See also
"Energy band"