Birefringence is a difference in refractive index that occurs along different axes in a material. In crystals like calcite this is an intrinsic effect resulting from the atomic structure. The result is that light travelling along one axis experiences a different refractive index than light travelling along a different axis.
In addition to atomic structure there are other physical phenomena that cause birefringence in materials. For example in polymeric materials like plastics the molecular structure is normally isotropic. However, manufacturing processes, such as extrusion and injection molding, create anisotropies in the material which lead to physical stress and optical birefringence.