Magnetism is a physical phenomenon. All fundamental charged elementary particles have a characteristic magnetic moment, which add up additively from the contributions of the individual building blocks in a solid.
Diamagnetic materials develop an induced magnetic field in an external magnetic field in a direction opposite to the external magnetic field.
Paramagnets are substances that are attracted by a magnet and follow the external field in their magnetization. Without an external magnetic field, paramagnetic materials show no magnetic order.
The respective neighboring elementary magnets carry the same magnetic moment, but their orientation is opposite to each other (antiparallel). Antiferromagnetically ordered materials do not exhibit an external permanent magnetic moment.
Ferromagnets either generate a permanent magnetic field themselves or are strongly attracted by a pole of an external magnetic field. The magnetic moments of the atoms of the material tend to align parallel.
Nonmagnetic elements are substances without magnetic properties. They do not have their own magnetic field and cannot be influenced by external fields.