tetrahedral and octahedral holes
Stack oranges as tightly as you can and you still cannot fill all the space — there are gaps between them. The same is true when atoms close-pack: 26 percent of the volume is empty, and that empty space is not random. It comes in two distinct kinds of hole, defined by the shape made by the spheres surrounding each gap. These holes are where the smaller ions of an ionic crystal go, so understanding them is the key to reading almost any inorganic structure.
A tetrahedral hole is the small gap where one sphere of one layer rests in the dimple of three spheres of the next: four spheres surround it, sitting at the corners of a tetrahedron, so an ion placed there has coordination number 4. An octahedral hole is larger — it is bordered by three spheres in one layer and three in the next, six in all, arranged at the corners of an octahedron, giving coordination number 6. The counting is clean and worth memorizing: for every N close-packed spheres there are exactly N octahedral holes and 2N tetrahedral holes. The octahedral holes are bigger (a sphere of radius up to 0.414 times the host radius just fits) than the tetrahedral ones (up to 0.225 times).
These holes generate the whole family of ionic structures. If anions close-pack and cations fill all the octahedral holes, you get rock salt (NaCl). Fill all the tetrahedral holes and you get the antifluorite arrangement; fill half of them and you get zinc blende or wurtzite. Fill half the octahedral holes and you get rutile or cadmium-chloride layers. Whether a given cation chooses a tetrahedral or octahedral hole is governed largely by its size relative to the anion — the radius-ratio idea. So 'which holes, and how many are filled' is the single most useful question to ask of any close-packed ionic solid.
In a ccp array of 100 oxide ions there are exactly 100 octahedral holes and 200 tetrahedral holes. In the mineral spinel, MgAl2O4, the magnesium ions sit in tetrahedral holes and the aluminium ions in octahedral holes of just such an oxide framework — only a fraction of each kind of hole is filled.
Counting holes (N octahedral, 2N tetrahedral per N spheres) and which are filled generates the common structure types.
A common slip: there are twice as many tetrahedral holes as octahedral, but the tetrahedral holes are smaller. Filling all the smaller holes (antifluorite) requires the cations to be small enough to fit a coordination of only 4.