Electromagnetic induction. A changing magnetic flux induces a current into a coil.
UPDATE (14/05/2011): An extended, 15-minute video on the topic is now available at http://www.youtube.com/watch?v=l-RjuauyuzM or on my website at www.ibphysicshelp.net.
Demo 1: As the magnet is moved in, the magnetic flux through the solenoid changes and an induced current appears (Faraday's law). The faster the magnet the higher the induced current. If the solenoid is approached first with the other magnetic pole, the direction of the induced current changes. When the magnet is moved away from the solenoid the direction of the current changes again.
Demo 2: Same as above but using a different coil and a digital multimeter.
Demo 3: The set up consists of two solenoids: one large one connected in a simple circuit and a second, smaller one, connected to an ammeter. When the switch is closed, a DC current is established in the circuit and a steady magnetic field is produced in the large solenoid. There is no induced current in the small solenoid as the magnetic flux through it does not change. However, when the switch is switched on or off, an induced current is produced. This is because for a short period of time the current changes and therefore the magnetic field produced by the large solenoid changes as well - hence an induced current in the small solenoid.
DOWNLOAD: An iPod/iPhone version of this video is available for download at: http://traffic.libsyn.com/ibphysicshelp/Electromagnetic_induction_iPhone__iPod.m4v