When individual atoms join to form a solid crystal, their atomic orbitals merge together. This merging happens because of quantum mechanical interactions between the atoms. Instead of separate energy levels, the electrons share a massive spread of energy levels called bands.
The two most important bands are the valence band and the conduction band. The valence band holds the electrons that participate in chemical bonding. The conduction band sits above it and is mostly empty.
If electrons gain energy, they can jump into the conduction band to move freely. This movement of electrons is what creates electrical current. The gap between these two bands is called the band gap.
A large gap means the material is an insulator, like glass. A small gap makes it a semiconductor, like silicon. No gap at all makes it a conductor, like copper.
Students often think that electrons in solids belong to single atoms. In reality, these valence electrons are shared across the entire solid material. You can think of the band gap as a fence.
Insulators have a tall fence that electrons cannot climb. Conductors have no fence at all, so electrons roam freely. Semiconductors have a short fence that can be crossed under the right conditions.