You will have to do a little reading to fully understand it if you haven't yet seen the tensor representation of crystal properties and optical properties in particular. But the essential point is that refractive index can vary for different polarizations in a material. For a uniaxial crystal, which is the most typical case, it turns out that you can represent the refractive index by drawing an ellipsoid with a=b =/= c if a,b,c are the three semimajor axes. Then if you draw a vector from the center to any point on the surface, that length will be inversely related to the value of the refractive index felt for a wave with polarization pointing in that direction. So, the optic axis of the material is the long axis of this ellipse, and if the wave is propagating along the optic axis, it will feel the same refractive index regardless of polarization. Therefore there is no birefringence experienced by the wave.
For any other direction of propagation, the refractive index will depend on the polarization state (draw a vector in the direction of propagation, and draw the ellipse normal to it which intersects the ellipsoid). In this case, two orthogonal polarization components will experience two refractive indices - called the ordinary and extraordinary index. If the wave impinges on the crystal at an angle, the different refractive indices will cause the two polarization components to refract at different angles, leading to two spatially separated beams, the visible manifestation of birefringence.
Again, you'll have to read through the math to see why the refractive index can be represented by this ellipse. It's not difficult but you just have to go through it.
So if one central intersection of the ellipsoid is a circle, the section orthogonal to that is the principal section. That's just terminology, I had to look that up as well. But the principal section will tell you the wave vector which experiences the most pronounced birefringence.
