What is Lattice, motif, unit cell?

A lattice is an infinite set of points, the nodes, obtained from one of them by all the translations $u\,\vec a + v\,\vec b + w\,\vec c$ with integer $u$, $v$, $w$. The crystal is obtained by placing on every node the same motif (one atom, a group of atoms or ions). A unit cell is a parallelepiped built on $\vec a$, $\vec b$, $\vec c$ whose translations fill space; its edge lengths and angles are the lattice parameters $a$, $b$, $c$, $\alpha$, $\beta$, $\gamma$. In a cubic cell $a = b = c$ and the angles are $90^\circ$.