In almost every book about solid state theory, electrons in a periodic potential are introduced and calculations for the dispersion relation $E(k)$ are presented. One obtains the usual pictures in reciprocal space, where the parabolas for the free electron gas are sketched and the impact of a weak periodic potential is displayed by band gaps at the edges of the first Brillouin zone.
One can see the Fermi surfaces for the cases where no potential is applied (left) and where a weak periodic potential is added. The fermi surface, which was a sphere before, now bends a little bit at the edges of the BZ. In the two sketches on the right side of the image, the first and second BZ are drawn. However, they are labeled with first and second band.
I also heard someone saying that the Brillouin zones equal the energy bands. As I understood it, the BZs are more a geometric construction and I cannot see how they should equal bands.
So the question is: What is the exact relation between energy bands and the Brillouin zones?