An optic axis of a crystal is a direction in which a ray of transmitted light suffers no birefringence (double refraction).
The internal structure of crystals (the specific structure of the crystal lattice, and the specific atoms or molecules of which it is composed) causes the speed of light in the material, and therefore the material's refractive index, to depend on both the light's direction of propagation and its polarization.
The refractive index of the ordinary ray is constant for any direction in the crystal.
The refractive index of the extraordinary ray varies depending on its direction.
It is the space and time average of the orientation of the long molecular axis within a small volume element of material demonstrating a mesophase.