The homopolar motor was the first electrical motor to be built. Its operation was demonstrated by Michael Faraday in 1821 at theRoyal Institution in London. Back then Michael used a bath of mercury and some pretty fancy equipment. The development of very strong magnets more recently has made it possible to recreate these motors at home. The key to understanding how they work is that idea that when a current passes through a wire (in this case the copper wire connected between the two ends of the cell) it generates a magnetic field. When this field is in another field generated (in this case) by a permanent magnet the wire is pushed.
Brace yourself here's a fuller explanation from Wikipedia...
The homopolar motor is driven by the Lorentz force: as it moves through a magnetic field, the conductor is pushed through a magnetic field by opposing forces. This force induces a torque around the axis of rotation. Because the axis of rotation is parallel to the magnetic field, and the opposing magnetic fields do not change polarity, no commutation is required for the conductor to keep turning. This simplicity is achieved at the cost of not being able to have more than one coil turn, which makes this configured homopolar motor unsuited for most useful applications. Homopolar motors have advantages and disadvantages and have not been fully developed.
How to make a homopolar motor:
First, be careful...
If two neodymium magnets are swallowed they can cause serious injury
Rechargeable batteries can explore when shorted, so only use non rechargeables.
The easiest to make are the coil and heart motors.
You will need:
Place the negative end of the cell onto the neodymium magnet
Bend the wire so that one end rests on the positive end of the cell and the other just about touches the magnet at the bottom.
If it doesn't move try tinkering with the copper wire shape. You could also try flipping the magnet over as the motor will only work with the magnet one way around.
The rolling one is much more difficult to get working:
You will need:
Place the neodymium magnets on each end of the battery. They should be arranged so that similar poles face each other.
Strip 20mm of insulation from each end of the wire shape the wire into a curve and curve the stripped ends so that they hook over each of the magnets.
The spare cell is because I found that by the time I'd tinkered enough to get the roller working I'd flattened the cell and needed to use a fresh one.