Permanent magnet attraction from Maxwell's equations / Lorentz force Edit: My question is different because I am not asking how permanent magnets come about. Rather, I am asking why (taking permanent magnets as a given) they attract or repel each other. I.e. What equation describes the force that causes a permanent magnet to stick to my fridge.
I have been searching online for some qualitative explanation of why permanent magnets attract each other (for example fridge magnets) and can't find anything.  The closest I can find is attraction/repulsion of two parallel current carrying wires.
Here's using the same steps to two permanent magnets situated on top of one another:
Lorentz force law says force is proportional to $\vec{v} \times \vec{B}$. Each magnet is composed of many atoms all oriented the same way. All together the electrons orbiting the individual nuclei sum to create a stronger magnetic field (let's say pointing up to the other magnet).
This other magnet is also composed of individual, similarly oriented atoms.  Since I know it will attract when the "south" pole of the top magnet is above the "north" pole of the bottom magnet, let's try that situation.
Trying to apply the Lorentz force law, $v$ is going around in a circle on the top magnet. $B$ is coming straight up (from the first magnet). This seems to imply that the force would be perpendicular to the $B$ field, but spinning around in time (as the electron spins around the nucleus).  
Where does the force that attracts the two permanent magnets towards one another come from?
 A: Permanent magnets can't be explained in terms of Maxwell's Equations or Classical Physics in general. As most (if not all) of the phenomena that arrises due to microscopic behaviour, you need Quantum Physics to explain it properly. 
Permanent magnetism comes from spin. Spin is an intrinsic property of every particle, like mass. It is quantized, coming in integer multiples of one half. It is one type of angular momentum. The important thing you need to know about spin, in order to answer your question, is that it couples with the Magnetic Field and also 'produces' its own little magnetic field. More precisely, the spin of a particle implies in a magnetic dipole moment. 
Just as every fundamental particle (like electrons and protons) has its spin, so do composite particles (like an atom). It happens, though, that the magnetic moment of a single atom is very weak, so you need all the atoms in a lattice to align, so that their fields sum up, in order to get something apreciable and detectable. And that's what happens when you put a non-magnetized piece of iron on a magnetic field. Its atoms align and you get a magnet, simmilarly to when you have lots of tiny magnets togheter.
But, now, why aren't all materials magnetic, you ask?
Because for most atoms the spins of different electrons, togheter with the spin of the nucleus, cancel out. You need a very specific arrangment of atomic structure in order to form crystals that can have its atoms with sobressalent spin and aligned, thus creating a magnet.
A: 
The closest I can find is attraction/repulsion of two parallel current
  carrying wires

.
There you go!
You can model the permanent magnet as made up of lots of little current loops that each form the dipole element of an atom. (Why that’s a good model is a separate question). Together they act like a bigger loop. 
The field from that also induces a corresponding magnet in the iron of the refrigerator door. Again, that looks like a current loop. 
And those two effective loops are parallel, so they attract. 
