the motif
On patterned wallpaper, the repeating picture — say a flower — is stamped at every grid point. In a crystal that repeating unit is the motif (also called the basis): the specific atom or group of atoms attached to each lattice point. The lattice decides where to stamp; the motif decides what is stamped.
The motif can be a single atom (copper: one Cu per point) or many atoms (a protein crystal: thousands). Every lattice point carries an identical copy, in the same orientation. For example, in the diamond structure the lattice is face-centered cubic and the motif is two carbon atoms, one at (0,0,0) and one at (1/4,1/4,1/4); so the repeated picture is that pair of carbons.
The motif is described by fractional coordinates relative to the cell, and we take the smallest set of atoms that, when copied by the lattice, reproduces the whole crystal (closely related to the asymmetric unit once symmetry is included). Do not confuse the number of atoms in the motif with the number of atoms in the whole unit cell.
Graphene's motif is two carbon atoms, and its lattice is a 2D hexagonal (triangular) lattice — hang that two-atom motif on every lattice point and the honeycomb sheet appears. The honeycomb itself is NOT a lattice (its two carbons have different environments); the lattice is the underlying triangular grid.
Motif (2 carbons) + lattice (triangular) = the graphene honeycomb.
The number of atoms in the motif is fixed by the structure, but where you place the motif's origin relative to the lattice point is a free choice — moving it just shifts all coordinates by a constant.