the band gap
The band gap is the forbidden zone in a solid's energy spectrum: a range of energies in which there are simply no electron states to occupy, sitting between the highest filled band and the lowest empty one. It is the single number that most sharply divides the material world into conductors, semiconductors, and insulators. Small gap, and heat or light can lift electrons across it; large gap, and they stay stuck.
Precisely, the band gap E_g is the energy separation between the top of the valence band and the bottom of the conduction band. It comes in two flavours: a direct gap, where both band extrema occur at the same wavevector k (as in gallium arsenide, ideal for LEDs and lasers because an electron can drop across the gap by emitting a single photon), and an indirect gap, where they lie at different k (as in silicon, where the transition also needs a phonon to conserve crystal momentum, making light emission inefficient). By rough convention a semiconductor has E_g of order 0.1 to 3 eV, an insulator larger (diamond about 5.5 eV), and a metal none at all. Microscopically the gap opens because Bragg reflection at the Brillouin zone boundary mixes plane waves, splitting the free-electron band by roughly twice the relevant Fourier component of the potential, 2|V_G|.
The band gap governs which photons a material absorbs or emits, and hence the colour of a semiconductor, the operating wavelength of an LED or laser, and the efficiency of a solar cell; it also sets the exponential temperature dependence of intrinsic conductivity through the factor exp(-E_g / 2 k_B T). The honest caveat is that there is no sharp dividing line between a semiconductor and an insulator; the distinction is a soft matter of gap size and of whether doping can populate the bands at accessible temperatures.
A red LED uses a semiconductor with a gap near 1.9 eV, since a photon of that energy has a wavelength around 650 nm; a wider-gap material such as gallium nitride, about 3.4 eV, emits blue or ultraviolet light instead.
The gap sets the minimum photon energy the material can emit or absorb across the bands.
There is no crisp line between semiconductor and insulator; both have a gap, and which name applies depends on gap size and whether carriers can be introduced thermally or by doping.