Magnet
Rare earth (lanthanoid) elements have a partially occupied "f" electron shell (which can accommodate up to 14 electrons). The spin of these electrons can be aligned, resulting in very strong magnetic fields, and therefore, these elements are used in compact high-strength magnets where their higher price is not a concern. The most common types of rare-earth magnets are samarium-cobalt and neodymium-iron-boron (NIB) magnets.
Nanomagnet
Canonical examples of nanomagnets are grains of ferromagnetic metals (iron, cobalt, and nickel) and single-molecule magnets. The vast majority of nanomagnets feature transition metal (titanium, vanadium, chromium, manganese, iron, cobalt or nickel) or rare earth (Gadolinium, Europium, Erbium) magnetic atoms.