What are meant by Bloch states and the Brillouin zone for a quantum mechanical particle moving in a one-dimensional periodic potential?

Derive an approximate value for the lowest-lying energy gap for the Schrödinger equation

$$-\frac{d^{2} \psi}{d x^{2}}-V_{0}(\cos x+\cos 2 x) \psi=E \psi$$

when $V_{0}$ is small and positive.

Estimate the width of this gap in the case that $V_{0}$ is large and positive.