the unit cell
The unit cell is the single stamp that, repeated over and over with no gaps and no overlaps, tiles all of space to build the crystal. Like one tile in an endless bathroom floor: know the tile and how it repeats and you know the whole floor. It is the smallest chunk that still carries the full repeating pattern.
Geometrically it is a parallelepiped (a slanted box) defined by three edge vectors a, b, c meeting at one corner, with edge lengths a, b, c and inter-axial angles alpha, beta, gamma. Stacking translated copies fills space. For example, a cubic cell is a plain box of side a, and copies stacked along x, y, z reproduce the crystal. The atoms inside, given by fractional coordinates, are the motif.
The unit cell is a choice, not a unique object — you can pick different cells that all tile the same lattice. By convention we pick one that shows the symmetry clearly, even when it is not the smallest. Also, atoms lying on cell faces, edges, and corners are shared with neighbouring cells, which matters when you count how many atoms belong to one cell.
Copper's conventional unit cell is a cube of side a = 3.615 angstrom containing 4 Cu atoms (at the corners and face centres). Stack this cube edge-to-edge in all three directions and you rebuild any amount of copper metal exactly.
One cube, tiled in 3D, is the whole crystal.
There is no single correct unit cell — infinitely many choices tile the same lattice. What is fixed is the lattice and the structure; the cell is just the box we draw around one repeat unit.