Born in Zonnemaire, Netherlands, Zeeman was a student of Lorentz at the University of Leiden. He began lecturing at the University Leiden in 1890. In 1896, at the request of Lorentz, he began investigating the effect of magnetic fields on a light source and discovered what is now known as the Zeeman effect. This discovery proved Lorentz theory of electromagnetic radiation.
Zeeman was appointed professor of physics at the University of Amsterdam in 1900. In 1908 he became director of its Physical Institute. He remained there until his death conducting research on the propagation of light in moving media such as water, quartz, and flint.
You can find a more detailed biography at the Nobel e-museum.