Birefringence is the difference between the highest and lowest refractive indices
Now to explain the above:
When light enters a crystal it is split into two rays that vibrate perpendicular to eachother, along an axis of propagation. In uniaxial crystals there is only one such axis. In biaxial crystals there are two of these axes, which make them more complicated, but the idea is the same.
The two rays the light is split into may have different velocities, in which case the mineral is said to be birefringent. When the light exits the mineral, it is re-combined to form a new light wave, a vector combination of the fast and slow rays. If the velocities of the two rays inside the crystal were equal, the light is unchanged.
Typically observations of this phenomenon are made using a thin section (30-50 microns) viewed with a petrographic microscope, under cross polarised light. When the polarizers are crossed, no unchanged light will be let through. This means that you will see black for a non-birefringent mineral. Birefringent minerals will show light, which may be any where from grey to neon, called interference colours.
The wavelength of the light can be determined by comparing the colour(s) viewed with an interference colour chart. This will tell of the difference between the velocities of the rays of light inside the crystal (called retardation).
The colour of the light will change upon rotation of the thin section. This is because the orientation of the axes along which the light propagates are moving along with the mineral. The birefringence will drop to zero every 90 degrees upon rotation of the mineral. This is called extinction. It occurs when the axes of light propagation are aligned such that the light exiting the crystal is aligned with the polarizer.
The first time I viewed this phenomenon, my thoughts were "this must be great when you're on drugs," because it can be very psychedelic. For some rock thin sections, it's like looking through a kaleidoscope as you turn the stage, with minerals lighting up and going extinct. I suggest you take a look if you are ever given the opportunity. Rocks and minerals aren't so boring after all!