Magnetic field causes a force to act on moving charges as in this equation: $F = qv*B$. So what is the force that causes a magnet to attract a charge-less piece of iron? or causes the like poles of two magnets to repel each other?
It is the magnetic force between magnets. It is very similar to magnetic force between electric circuits carrying electric currents, where this force can be explained as consequence of the Lorentz force $q\mathbf v\times \mathbf B$. One can model the magnet either as a set of current loops (the Ampere magnetic model) or a set of magnetic dipoles (the Coulomb magnetic model). In either case, the force between magnets is basic force of nature and we call it magnetic force.