Zeeman effect is the splitting of the energy levels in a system due to an applied (or effective - in the case of SO interaction) magnetic field.
Normally, all the energy levels are spin-degenerate in a molecule. And the up-spin and down-spin levels respond differently to a magnetic field.
This splitting is given by:
[tex] \Delta E = \mu_B \cdot \overline{B}[/tex]
where mu_b is the Bohr magneton.
You may want to check Stern-Gerlach experiment and Dirac equation if you want to delve into the exact formulation. I guess that's where it's fundamentally derived.