the space lattice
Imagine an endless three-dimensional wallpaper: space filled with identical dots, and around every single dot the view looks exactly the same. The space lattice is precisely this piece of pure geometry — an infinite array of points where, standing on any one point, the arrangement of all the other points looks identical to the view from every other point. There are no atoms yet, only geometry.
Formally, a lattice is the set of all points you can reach from one starting point using integer combinations of three non-coplanar translation vectors a, b, c: R = u a + v b + w c, where u, v, w are any whole numbers. Because every point is generated the same way, all lattice points are equivalent — you cannot tell one from another. In three-dimensional space there are exactly 14 distinct lattices (the Bravais lattices).
The lattice is not the crystal — this is the single most common beginner trap. The lattice is just a mathematical skeleton of points; the real atoms are the motif you hang on each point. Confuse the lattice with the crystal structure and your atom-counting will go wrong. A lattice point does not necessarily have an atom sitting on it.
Stand on any point of a simple cubic lattice and look around: you see six nearest points, one along each of +x, -x, +y, -y, +z, -z, all at distance a. Now teleport to a different lattice point and look again — you see exactly the same six-point view. That indistinguishability is the whole definition of a lattice.
Every lattice point has an identical environment — that is what makes it a lattice.
A lattice has no edges and no atoms — it is an infinite, purely geometric set of points. Real crystals are finite and made of atoms; the lattice is the idealized periodic scaffold we lay over them.