Atoms behave as magnets for two reasons.
First, the electrons which make up the atom are themselves
magnets, with magnetic dipole moments of magnitude
one Bohr magneton 
:
(Protons and neutrons also have magnetic moments, but they
are much smaller in magnitude than .)
Second, the atoms are ``orbiting'' the nucleus, and
this orbital motion is often equivalent to circulation
of charge, which of course means the electron is like
a current loop.
Hence, there is the possibility of an orbital dipole
moment as well.
These orbital dipole moments have magnitudes on the order
of a few Bohr magnetons.
Different materials respond to applied magnetic fields in
different ways because of the various ways the atomic
dipole moments respond to the applied field and to
the fields of neighboring atoms.