the space lattice
Strip away the atoms from a crystal and keep only an infinite array of identical points, each point seeing exactly the same surroundings as every other — like the intersection dots on infinite graph paper, extended into three dimensions. That skeleton of points is the space lattice. It captures the pure geometry of the repeat, before you say what actually sits at each spot.
Formally, a lattice is a mathematical array of points, and a crystal equals a lattice plus a basis (the single atom or the little group of atoms placed at each point). The points are generated by three vectors a, b, c: any lattice point is u times a plus v times b plus w times c, for whole numbers u, v, w. So the lattice tells you the rhythm of the repeat, and the basis tells you what is being repeated.
Separating where the repeat happens (the lattice) from what is repeated (the basis) is a powerful idea: it lets us classify every crystal on Earth using just fourteen distinct lattices. A single lattice point need not be one atom — in rock salt it stands for a sodium-chlorine pair, and in more complex crystals for a whole cluster.
Imagine an infinite 3D array of dots where each dot has an identical view of its neighbours. Put one copper atom on every dot and you get copper's FCC crystal; put a two-atom sodium-chlorine pair on every dot and you get rock salt. The dots (the lattice) stay the same; only the decoration (the basis) changes.
One lattice, many crystals — the basis is what you hang on each point.
A lattice point is not necessarily an atom; it is an abstract point of identical environment. A common beginner slip is to assume each lattice point equals exactly one atom.