You have presumably looked it (birefringence) up already and you need an explanation of the actual mechanism rather than a simple definition.
It occurs in a medium that is not isotropic - the refractive index along one axis is different from the refractive index along another axis. Refractive index is affected by how the molecules polarise in the presence of an electric field, which, in turn, affects the speed of the wave propagation through the medium. If the medium is a crystal or some 'stressed' amorphous solids, the electrons may be a bit more easily displaced in one direction than in another so the molecules can distort by a different amount. Calcite is the classic example of such a crystal. Light of one plane of linear polarisation travels faster than light in an orthogonal plane and so unpolarised light is split into two beams when it travels through obliquely - the random selection of different polarisation vectors from all the different waves passing through are all resolved into ('H' and 'V') components, corresponding to the two planes in the crystal, each one being refracted by a different amount.