A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. It consists of the product of a plane wave and a periodic function, which has the same periodicity as the potential. The plane wave wavevector is unique only up to a reciprocal lattice vector, which gives rise to the concept of the Brillouin zone. Generally, there are a number of solutions to Schrodinger's equation for a Bloch electron. They have a finite energy spacing. The bandstructure is the collection of energy eigenstates within the first Brillouin zone. All the properties of electrons in a periodic potential can be calculated from this bandstructure, at least within the independent electron approximation.
It can be shown that a particle in a periodic potential must have this form by proving that translation operators commute with the Hamiltonian.
