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Band theory classifies solids by the filling of their energy bands. In metals, the Fermi level lies within a band, so there are empty states immediately above E_F available for conduction. In insulators, all bands below a large gap (> ~4 eV) are completely filled and all above are empty — no states are available at accessible energies. Semiconductors are insulators with small gaps (< ~3 eV) where thermal excitation or doping can promote electrons across the gap, creating mobile carriers. This classification explains why copper conducts, diamond does not, and silicon can be made to do either.
The most consequential prediction of band theory is the division of crystalline solids into three categories based on how their energy bands are filled. In a metal, the Fermi level cuts through one or more bands, leaving partially filled states at the Fermi energy. These electrons can be accelerated by an arbitrarily small electric field, producing electrical conduction. The high density of states at E_F also gives metals their characteristic large electronic specific heat and Pauli paramagnetism.
In an insulator, all occupied bands are completely filled and separated from the empty bands by a large energy gap E_g. Since a completely filled band carries no net current (for every electron moving right, there is one moving left), an applied field cannot accelerate the electrons — you would need to excite an electron across the gap. For diamond (E_g = 5.5 eV), room-temperature thermal energy k_BT ~ 0.025 eV is utterly inadequate to excite any appreciable number of electrons across the gap, so the conductivity is essentially zero.
Semiconductors are the intermediate case: the gap is small enough (roughly 0.1 to 3 eV) that some electrons are thermally excited across it at room temperature, or the gap can be overcome by doping. The intrinsic carrier density scales as n_i proportional to exp(-E_g / 2k_BT), which is exponentially sensitive to both the gap size and the temperature. For silicon at 300K, n_i is approximately 1010 cm-3 — small compared to a metal's 1023, but enough to make silicon a useful conductor under the right conditions. The ability to control this carrier density through doping is what makes semiconductors the foundation of modern electronics.
The boundary between "insulator" and "semiconductor" is not sharp — it is a matter of gap size and practical utility rather than a fundamental physical distinction. Materials with gaps larger than about 3-4 eV are usually called insulators, smaller gaps semiconductors. Semimetals (like bismuth and graphite) represent a fourth category where the gap is actually negative: bands overlap slightly, creating small pockets of both electrons and holes even at zero temperature. The richness of this classification — and its exceptions, including Mott insulators where strong electron-electron interactions open gaps that band theory misses — is what makes condensed matter physics endlessly interesting.