N.P. Armitage notes on Zeeman experiment

The Zeeman effect is the observed splitting of a well defined spectral resonance into several multiplets in the presence of a magnetic field.

In the vast majority of atoms, there exist several electron configurations with the same energy i.e. they are degenerate, so that transitions between different sets of configurations correspond to a single excitation lines.
The existence of a magnetic field breaks this equivalency i.e. its breaks the degeneracy, since the field interacts in different ways with electrons with different quantum numbers.  The result is that, where as before there were several configurations of the same energy, now their energies are split, which gives rise to several very close spectral lines.
 

The Zeeman effect is important in various applications such as electron spin resonance and magnetic resonance imaging i.e. NMR which you have already performed

Pieter Zeeman

Useful Links:

Leybold's description of their Zeeman effect experiment  --> Quite explanatory

Wikipedia's entry on the Zeeman effect

Wikipedia's entry on the 2nd Nobel prize winner ever Pieter Zeeman