Short intro to symmetry, crystal systems and Bravais lattices

Useful page: Physics in a nutshell

Direct lattice (Bravais lattice):

$\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3 $, where $n_i$ are any integers and $\mathbf{a}_i$ are the primitive translation vectors (primitive lattice vectors) which lie in different directions and span the lattice.

Read more here and here http://www.physics-in-a-nutshell.com/article/6

Reciprocal lattice:

$\mathbf{G} = n_1 \mathbf{b}_1 + n_2 \mathbf{b}_2 + n_3 \mathbf{b}_3 $, where $\mathbf{b}_i$ are the three reciprocal primitive vectors and satisfy:

$\mathbf{a}_i \mathbf{b}_j = 2\pi \, \delta_{ij}$, and $i,j = 1,2,3$.

Read more here

Brillouin Zone construction

see animation here

and see 3D examples here

Triangular lattice

Honeycomb lattice

For fun see