Basically, the source of magnetism is a motion of charged particles, e.g., electric current induces magnetic fields. From a view of quantum physics, atoms in materials, e.g., solids, have spin, angular momentum quantum numbers. These are the magnetic states of the materials. Spin does not have any correspondence to classical concepts but angular momentum has. For example, a circular motion of electron in an atom makes electric current which induces magnetic field. You may consider each atom in the material as a tiny magnet. It is very strong in some atoms. Good magnetic materials have a nice alignment of the atomic magnet. In some materials, the alignement is random, in particular, at high temperature.
Anyway, if you want ot know other problems associated with magnetic properties in depth, you need to understand Maxwell's equations.
