the ionic radius
When an atom becomes an ion, losing or gaining electrons, its size changes noticeably, and we give that new size its own name: the ionic radius. Lose electrons to form a positive ion (cation) and it shrinks; gain electrons to form a negative ion (anion) and it swells. A sodium atom loses one electron and its whole outer shell empties, so Na^+ is much smaller than neutral sodium.
The reasons are concrete. A cation shrinks because it has fewer electrons for the same nuclear pull (and often an entire outer shell is gone): Na is about 1.9 angstrom but Na^+ is about 1.0 angstrom, nearly half. An anion swells because extra electrons crowd in and repel each other while the same nucleus holds them: Cl is about 1.0 angstrom but Cl^- is about 1.8 angstrom. So in an ionic solid the anions are usually the big spheres and the cations the small ones tucked between them.
Ionic radii govern ionic crystal structures directly, through the radius ratio, the cation radius divided by the anion radius. That ratio decides how many anions can pack around each cation (its coordination number) and therefore which structure forms: a small cation fits only a few large anions around it, a larger one fits more. This is why sodium chloride, caesium chloride, and zinc blende, all simple 1-to-1 ionic solids, nevertheless adopt three different structures. Get the ionic radii and the radius ratio, and you can often predict the structure on the back of an envelope.
In NaCl the radius ratio is about 1.0/1.8 = 0.56, which favours each Na^+ sitting in an octahedral hole surrounded by 6 Cl^- (coordination 6). Swap in the larger Cs^+ (radius about 1.7 angstrom) and the ratio rises above 0.73, so CsCl takes 8-fold coordination instead, a different structure entirely, driven purely by size.
The ratio of ionic radii sets the coordination number and hence the crystal structure.
Ionic radii are not absolute either; they come from splitting a measured cation-anion distance into two parts, and that split depends on the reference chosen (Pauling's and Shannon's tables differ). They also change with coordination number, so the same ion has slightly different tabulated radii in different structures.