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In most atoms, there exist several electronic configurations that have the same energy, so that transitions between different pairs of configurations correspond to a single line.
The presence of a magnetic field breaks the degeneracy, since it interacts in a different way with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, now there are different energies, that give rise to several very close spectral lines.

Without a magnetic field, configurations a, b and c have the same energy, as do d, e and f. The presence of a magnetic field splits the energy levels. A line produced by a transition from a, b or c to d, e or f now will be several lines between different combinations of a, b, c and d, e, f. Not all transitions will be possible -- see transition rules.
Since the distance between the Zeeman sub-levels is proportional with the magnetic field, this effect was used by astronomers to measure the magnetic field of the Sun and other stars.
There is also an "anomalous Zeeman" effect that appears on atoms with odd atomic number, the number of Zeeman sub-levels being even instead of odd. This could be explained with the existence of an angular momentum that allows half-integer values.
The Zeeman effect is named after the Dutch physicist Pieter Zeeman.
Bibliography
See also: Stark effect