What is saturation in an inductor
Rawbrawb's answer doesn't explain the actual mechanism by which saturation occurs, which is a fairly easy to understand:

It helps to first understand how materials generate magnetic fields. A simple way to think of this is as each atom being a small loop of current which generates a magnetic field. 



A magnetic material has huge numbers of these loops. These loops tend to align themselves into "magnetic domains", which are microscopic areas where all of the loops are in alignment. In an unmagnetized material, the directions of the domains are randomly distributed, and so there is no net magnetic field. 



Applying a magnetic field to a ferromagnetic material will start to align the magnetic domains, resulting in an "induced" magnetic field from the material. Increasing the applied magnetic field will increase the amount that the magnetic domains are aligned, and so increase the induced magnetic field. This is typically very non-linear. At some point, the applied magnetic field aligns ALL of the domains, and it is no longer possible to increase the magnetic field from the material. This state is known as "saturation".