Means there are two axes with different refractive indices in the material. When a material is described as birefringent all it really means is that a light wave travelling through it can have two distinct velocities. In other words, the material has two distinct values of refractive index (another term for birefringence is ‘double-refraction’). Birefringence is the fundamental principle by which polarization maintaining or HiBi fiber works.
The term ‘birefringent axes‘ refers to the planes in which a light wave will be subjected to each of the two distinct values of refractive index - by definition, there can only be two birefringent axes. As the index along these will be different, the phase velocity of the light is different along the axes, and they are referred to as the slow (higher index) and fast (lower index) axes.