The lower energy group are called valence electrons (holding the solid together) and the higher group are, generally, free to move. They are called the conduction electrons.
Conductors are unusual because their two bands overlap. This means electrons can easily be promoted to the higher band and move, carrying a current.
In semiconductors, there is a narrow gap with only very few electrons in the higher conduction band. They thus allow only very little current to flow at room temp. Due to the small gap a small increase in temperature is thus sufficient to promote more electrons and so they conduct readily at elevated (50+ degrees for example.)
Insulators have two bands, but the gap between them is very large and no electrons are in the upper band. The temperatures required to promote an electron are typically sufficient to cause the material to disintegrate i.e. rip out the valence electrons holding it together.