Look straight down on a rhombus shaped calcite crystal (mineral specimen stores often have them) and an object beneath it is seen double. Rotate the crystal. One image stays nearly stationary and the other (the one farthest from the top blunt corner) moves around it. Calcite is 'birefringent' or 'double refracting'.
Calcite is a crystalline form of calcium carbonate, CaCO3, and the crystal is a lattice of positively charged calcium and negatively charged carbonate ions. The very large optical effects shown by calcite arise because the flat carbonate ions are arranged in parallel sheets. Light travelling perpendicular or parallel to them has very different electronic interactions. The crystal is anisotropic and its optical properties are direction dependent.
Light entering the crystal splits into two polarised components because the differently oriented electric vectors of the radiation interact differently with the induced dipoles of the carbonate oxygen.
One ray, called the ordinary or O ray, behaves in the familiar way of light passing through glass or water (as predicted by Snell's law of refraction). The second - the extraordinary or E ray - is indeed extraordinary, it is refracted through a different angle to the O ray. Worse, Snell's Law does not predict the refraction because its extent, given by the refractive index, is direction dependent.
At right, the upper unpolarised view shows the double image. In the lower views a plane polarising camera filter is rotated to reveal one or other of the polarised images.
All non-cubic symmetry crystals (calcite is trigonal, common salt is cubic as also is diamond) are birefringent but often only to an unnoticeably small degree. Ice (hexagonal symmetry) is birefringent enough that a carefully rotated polarising filter shows the presence of doubly imaged sundogs (see Richard Fleet's images).
The O and E rays passing through thin sheets of birefringent materials have different optical path lengths. The rays interfere giving the coloured bands that are often seen when looking through airplane windows.