A common core is Aluminum Nickel Cobalt (i.e., AlNiCo V) and a common coating is polypropylene. However, stirring bars can be manufactured out of many other materials, particularly for special purpose applications.
Other coating materials are polytetrafluoroethylene (i.e., PTFE, Teflon), ZedEx, and glass. Other core materials are rare earth magnets, such as Samarium Cobalt (i.e., SamCo) or Neodymium (i.e., NdFeB).
Magnetized cores are made of permanent magnets, meaning that they have their own magnetic fields and are able to retain their magnetic properties in the absence of an inducing field or current. Stirring bar cores are manufactured in a variety of ways, depending on the core material (e.g., production of Neodymium magnets, production of Aluminum Nickel Cobalt magnets).
The process of how commercially produced magnets get magnetized is also interesting (e.g., magnetizing ferromagnets).