You should be able to draw an approximate $E\text{ vs. }k$ dispersion relation for an electron moving in a one-dimensional potential. Your drawing should include the band gaps and Brillouin zone boundaries. There are propagating solutions in the bands and exponentially decaying solutions in the band gaps.
You should know that the wave functions for electrons in a periodic potential have Bloch form $\psi_{\vec{k}} = e^{i\vec{k}\cdot\vec{r}}u_{\vec{k}}(\vec{r})$. Here $u_{\vec{k}}(\vec{r})$ is a periodic function with the periodicity of the crystal. Wave functions with this form are eigenfunctions of the translation operator.
If the amplitude of the periodic potential that an electron in a metal experiences is small, the band diagram will resemble that of free-electron parabolas drawn in the reduced zone scheme. Small band gaps will appear at the Brillouin zone boundaries.
There are N allowed k-vectors in the first Brillouin zone where N is the number of unit cells in the crystal. Two electrons can occupy each $\vec{k}$ state, one with spin up and one with spin down.
The electron states that contribute to properties of metals, such as the specific heat or electrical conductivity, are close to the Fermi surface. For a metal, the Fermi surface separates the occupied states from the empty states in reciprocal space.
The Fermi surface for free electrons is a sphere of radius $k_F = ( 3 \pi^2 n )^{1/3}$, where $n$ is the electron density. When the electron density increases so that the Fermi surface comes close to the Brillouin zone boundaries, the Fermi surface bends to touch the Fermi surface at 90°.
Metal: The chemical potential is in a band so that there are empty electron states with energies just above the occupied electron states.
Semimetal: A metal with a low density of states at the Fermi energy.
Semiconductor: The chemical potential lies in a band gap and the band gap is smaller than 3 eV.
Insulator: The chemical potential lies in a band gap and the band gap is larger than 3 eV.
There are many methods for calculating electronic band structures. You should be able to find the band structure for some crystal online and be able to tell from the band structure and the density of states if that material is a metal, a semimetal, a semiconductor, or an insulator.
You should be able to explain the plane wave method and tight binding.
If you have a band structure relation $E(\vec{k})$, you should be able to describe how the density of states is calculated.
You should be able to describe how the electron contribution to the internal energy density is calculated from the electron density of states, and how other thermodynamic properties such as the specific heat, entropy, to Helmholtz free energy could be calculated from the internal energy.
