My second grader thought making a homopolar motor for her science experiment would be fun. And, it was. Now I am trying to explain how it works and the Lorentz force. Please help me by giving me a very simple explanation to what is turning out to a very complex theory (or law).
When you have current flowing in a conductor at right angles to magnetic field lines (or at some angle other than parallel, the conductor experiences a force at right angles to both of them. (It's described with the right-hand rule.)
Your Wikipedia link gives a very good explanation.
EDIT: let me see if I can draw what I think is going on in the cylindrical portion of the nail:
Here B is the magnetic field coming up through the nail and exiting horizontally out the side (and then bending back down toward the opposite pole of the magnet). I is the electric current, traveling along the length of the nail. Near the surface of the nail, I and B are at 90 degrees to each other. F is the resulting force, which goes in one direction all around the outside of the nail, causing it to spin. (I may have the polarity wrong.)