Explain how an electric motor works, including a discussion of the right-hand rule.

The Right Hand Rule is a useful mnemonic to spatially relate the force and velocity of a positively charged particle in a magnetic field.  Holding your hand as if you were going to shake someone else's, your thumb pointing up is the direction of velocity (conductor), your fingers pointing out is the direction of the magnetic field (flux), and your palm pointing to the left is the direction of the force (current). Force is thus perpendicular to velocity and magnetic field, or, if considering a motor, current is perpendicular to conductor and flux.

A motor works because 2 magnetic fields are interacting with each other, one from a fixed magnet and the other from an electromagnet.  The armature of a motor is an electromagnet with North and South polarities, and these interact with the North and South polarities on the magnet that encases the armature.  The North and South poles of the magnet repulse the North and South poles of the armature, causing the armature to spin. See graphic at the links: