Every metal you have ever used is a pile of spheres packed as tightly as the geometry allows. The densest packing of equal spheres fills seventy-four per cent of the space, and the arrangements that get close to it — face-centred cubic and hexagonal close packed — are the same arrangement with the same neighbours, seen from a different angle. The rest of the periodic table is what happens when you insist on something more symmetric.
Stack spheres in rows and the second layer sits in the hollows of the first — but there are two kinds of hollow, and only one of them is a proper fit. A layer sitting in the triangular hollows gives cubic close packing, which is the same thing as face-centred cubic seen from a different angle. A layer sitting in the cubic hollows gives hexagonal close packing, and it is marginally denser, which is why the densest packing of equal spheres fills 74.05 per cent of the space and not 74.048.
Everything less dense — body centred, simple cubic — is a structure that has given up a little packing in exchange for something else: a lattice with more symmetry, or a direction along which the atoms are evenly spaced.
The coordination number — how many neighbours touch a given atom — is the number that governs almost everything: iron is body centred with twelve neighbours and melts at a thousand degrees less than aluminium, which is face centred with twelve neighbours and a much stronger bond. Copper and diamond have the same coordination number and completely different structures, and diamond is hard because its bonds point in four fixed directions.
Everything a crystal does — the way it cleaves, the way it conducts, the way it bends light — comes out of the arrangement of atoms and the symmetry of the cell they sit in.