First of all electrons carry a single electronic charge and have a spin of 1/2. This implies that they are Fermions, particles which obey the Pauli exclusion principle. The Pauli exclusion principle states that no two Fermions can occupy the same quantum state described by a complete set of quantum numbers. We will have to simply accept these properties as they are based on experimental observation and can not be explained in any way.
Each quantum state has a specific energy associated with it. These energies are obtained as possible eigenvalues of Schrödinger's equation. The eigenvalues are the possible solutions to a system which is specified by the potential within the material and the mass of the particle.
Of all the possible energies we still have to find which one is the actual energy of a particle in question. This issue is further complicated by the fact that most practical systems do contain a large number of particles.
At absolute zero temperature the situation is somewhat easier since particles simply occupy the lowest possible energy (or state with the lowest energy). Since only one electron can occupy a specific state, the next electron will occupy the next state which can have the same or a higher energy.
Solids have a very large number of particles and possible energy levels. We will therefore consider groups of energy levels or energy bands rather than individual energies. Semiconductors are further characterized by the gap between the highest "almost-filled" band and the lowest "almost-empty" band. It turns out that this energy band gap plays a crucial role in the analysis of semiconductors.