Band theory or band structure describes the quantum-mechanical behavior of electrons in solids. Inside isolated atoms, electrons possess only certain discrete energies, which can be depicted in an energy-level diagram as a series of distinct lines. In a solid, where many atoms sit in close proximity, electrons are “shared.” The equivalent energy level diagram for the collective arrangement of atoms in a solid consists not of discrete levels, but of bands of levels representing nearly a continuum of energy values. In a solid, electrons normally occupy the lowest lying of the energy levels. In conducting solids the next higher energy level (above the highest filled level) is close enough in energy that transitions are allowed, facilitating flow of electrons in the form of a current. In insulating solids the next higher energy level lies far above the highest filled level (separated from it by an energy gap), prohibiting electrical current. Semiconductors are actually insulators, but their conduction is enough that they are classified separately. Their gap lies between those of conductors and insulators; the energy gap is small. (see gallery image 1)
Topological insulators are a class of insulators that have unique conduction properties. See gallery image 2 for their band structure and how it relates to the traditional band theory of solids.