the coordination number
The coordination number is simply the count of an atom's nearest neighbours, the atoms it directly touches or bonds to. Think of standing in a packed crowd and asking how many people you can reach out and hold hands with at once: that number is your coordination. In a crystal it captures how tightly and how symmetrically an atom is surrounded.
For close-packed metals (FCC and HCP) the coordination number is 12, the maximum for equal spheres. Body-centred cubic gives 8, simple cubic gives 6, and the open diamond structure gives just 4. In ionic crystals we usually mean the number of oppositely charged ions touching a given ion: sodium chloride puts each ion in a coordination of 6 (octahedral), caesium chloride gives 8 (cubic), and zinc blende gives 4 (tetrahedral). The higher the coordination, the more neighbours share the load and, generally, the denser the packing.
Coordination number ties together several ideas in this field. It is set largely by the radius ratio of the two ions and is the subject of Pauling's first rule; it distinguishes structure types (rock salt versus caesium chloride versus zinc blende are basically choices of coordination 6, 8, or 4); and it governs properties, since more bonds per atom usually means higher melting point and stiffness.
In table salt each Na+ is touched by 6 Cl- (and vice versa): coordination 6. Swap to CsCl and it rises to 8 because Cs+ is bigger.
Bigger ions can be surrounded by more neighbours, so radius ratio drives coordination.
Coordination counts only nearest neighbours. In BCC the eight body-diagonal neighbours count as coordination 8, even though six more sit just a little farther out; where exactly you draw the line is a convention.