Moonstone is a feldspar, generally an alkali feldspar in which two phases, typically orthoclase and albite, have separated into alternating microscopic layers as the crystal cooled slowly. Light entering the stone is scattered by these layers, and because the layers are of a thickness comparable to the wavelength of light, the scattering is preferential toward the blue end. The result is the floating blue-white sheen called adularescence, which appears to lie beneath the surface and to move as the stone is turned.
The name comes from Mount Adular in Switzerland, an old locality for the material. The best modern source has been Sri Lanka, with blue-sheen material from there and from Burma; rainbow moonstone, widely sold, is a labradorite rather than an orthoclase feldspar and is a different mineral marketed under the moonstone name.
The mechanism is the same class of phenomenon as the play of colour in opal and the schiller of labradorite: structural optics rather than pigment. Nothing in the stone is blue. This matters for the tradition, because the sheen's resemblance to moonlight is a genuine optical fact and the inference from it (that the stone is lunar, feminine, tidal, connected to cycles and to dreaming), is association by appearance. The resemblance is real; everything built on it is analogy.
Moonstone is soft for a ring stone at around 6 to 6.5, with two directions of perfect cleavage, and it chips readily. Feldspar is the most abundant mineral group in the earth's crust, so the material is common; only the sheen is uncommon.