What follows can be read in Demtröder "atomic phyiscs", keyword "Zeeman effect".
The B-field defines the quantization axis. That is, because the atom does not care, in which direction the photon travels, but just interacts with its electric field. Therefore all what matters to the atom is the direction the electric field of the photon oscillates.
Now, put the B-field along the z-axis and look at the emission proceess instead of the absorption. The atom has several ways to emit the photon.
1,2) into the z-direction (=axial direction) + circular polarized light. Depending on the rotation direction (clockwise or anti-clockwise) you call these photons sigma- or sigma+
3,4) into the radial direction + linear polarized light. Depending on the polarization the photon is either pi or sigma polarized. ONCE AGAIN: sigma (withou plus or minus) means LINEAR polarized.
It's not trivial, so you better check out the book.
Cheers.
semoi