the unit cell

Think of a single tile of wallpaper that, stamped over and over with no gaps, reproduces the whole wall. The unit cell is the crystal's version of that tile: the smallest box that, when stacked in all three directions without overlaps or holes, builds the entire crystal. Choose the cell, know the whole crystal.

A unit cell is described by its edge lengths a, b, c (the lattice parameters) and the angles alpha, beta, gamma between those edges. Atoms shared with neighbouring cells count only fractionally: a corner atom is shared by eight cells, so it counts one-eighth; a face atom is shared by two, counting one-half; an atom fully inside counts one. For the face-centered cubic cell, 8 corners times 1/8 plus 6 faces times 1/2 equals 1 plus 3, which is 4 whole atoms per cell.

From this one little box you can calculate density, packing efficiency, the number of atoms, and how to label planes and directions. The choice of cell is not unique — a primitive cell holds the bare minimum, while a larger conventional cell is often chosen because it displays the crystal's symmetry more clearly.