a crystal lattice
A crystal lattice is the invisible scaffolding of a crystal: an infinite pattern of points that repeats identically in every direction, like a three-dimensional wallpaper with no seams. Stand at any lattice point, look around, and the arrangement of everything you see is exactly the same as from any other lattice point. That perfect sameness under shifts is the single fact from which almost all of solid-state physics is deduced.
Precisely, a Bravais lattice is the set of all points R = n1 a1 + n2 a2 + n3 a3, where n1, n2, n3 are any integers and a1, a2, a3 are three fixed primitive vectors that are not coplanar. The defining property is translational symmetry: shifting the whole lattice by any vector R lands every point back onto a lattice point, so the crystal looks unchanged. The small region that tiles all of space when repeated by these vectors is a primitive cell; a real crystal is a Bravais lattice plus a basis, the group of one or more atoms attached identically to every lattice point. In three dimensions there are exactly 14 distinct Bravais lattices.
This periodicity is the foundation that makes solids tractable: it forces Bloch's theorem, defines the reciprocal lattice and Brillouin zone, and organizes electrons into energy bands. A common confusion to resist: the lattice is a mathematical set of points, not the atoms. The atoms sit at (or near) lattice points via the basis, and real crystals always depart from the ideal through surfaces, defects, impurities, and thermal vibrations.
Table salt (NaCl) is a face-centred cubic Bravais lattice with a two-atom basis, one Na+ and one Cl-; each ion sits on its own interpenetrating cubic pattern about 0.28 nm apart.
A crystal = a Bravais lattice (the repeating points) + a basis (the atoms at each point).
The lattice is a set of points, not the atoms; the atoms enter through the basis. A lattice with a one-atom basis and a lattice with a two-atom basis can share the same Bravais lattice yet be very different crystals.