Diamagnetic Materials
All matter is a little diamagnetic, but it is generally weak. It occurs when the non-cooperative electrons of these substances are exposed to a magnetic field. The atoms of diamagnetic substances have no net magnetic moments (meaning all the orbital shells are filled, so there are no unpaired electrons), but when they are exposed to a field, they produce magnetization. Quartz and water are both diamagnetic. You can actually bend water with a strong enough magnet.
Paramagnetic Materials
Some of the atoms or ions in these materials have a net magnetic moment (they have unpaired electrons in partially filled orbitals), but the individual magnetic moments do not combine. Like in diamagnetism, the magnetization is zero when there is no field, but in the presence of a field, they partially align, producing magnetism. Clay, sulfide, and silicate are all paramagnetic.
Ferromagnetic Materials
When you think of magnetic materials, you probably think of ferromagnetic materials, like iron, which is what the root "ferro-" refers to. In these materials, as in paramagnetic materials, atoms interact strongly because they have unpaired electrons, but unlike paramagnetic materials, ferromagnetic materials spontaneously create parallel or anti-parallel alignments of magnetic moments. They do not require the introduction of an external force field to align. This is a result of quantum properties of the electrons: namely, spin and the Pauli Exclusion Principle. Ferromagnets are the strongest magnets and are the only type that can be felt. Nickel and cobalt are also ferromagnetic materials. Many ordinary objects like dollar bills have traces of iron, which can make them attract magnets.