What exactly are rare-earth magnets and how are they formed?

Rare earths are the 14 elements in the periodic table that start with the element Lanthanum and end with the element Luthecium. The rare earth elements are magnetic because their 4f shells are not full. The elements as one moves to the right in the periodic table fill one electron at a time. The 4f shells each can have two spin states, up or down. Because it is more energetically favorable, Hunds Rule, the 1st 7 electrons fill with their spins all up. Spin state for Eu is around 7 times 1/2 (from each electron). Rare earth magnets are made from rare-earth elements. Typically these are compounds or composites with Fe.

The rare earths, also called the "lanthanides" have unpaired electrons in their "f" orbitals. Various compounds of these elements are used to produce very strong magnets. The particular compounds depends upon both chemical, engineering, and economic considerations. Search the term: "rare earth magnets" on the Internet and I think you will find a wealth of info -- probably more than you want to know.