Faraday's law and Lenz's law

In 1831, Faraday made a crucial discovery: a changing magnetic flux creates a voltage in a circuit — an electromotive force (e.m.f.) that sets the charges in motion, hence a current. This is electromagnetic induction. Note carefully: it is the variation of flux that matters, not the flux itself. A magnet held still near a coil produces nothing; a magnet you move does.

Faraday's law quantifies the induced e.m.f.:

   e = − dΦ/dt        (induced e.m.f. = rate of change of flux over time)

The faster the flux varies, the greater the e.m.f.: a magnet moved quickly induces more voltage than a slow one.

   magnet        coil
    [N|S] -->    (((|)))     magnet approaches : Φ increases -> induced current
                             magnet moves away : Φ decreases -> reversed current
                             magnet at rest : Φ constant -> NO current

What does the minus sign mean? It is Lenz's law: the induced current always flows in the direction that opposes the change that gave rise to it. If you bring a magnet closer, the induced current creates a field that repels the magnet; if you move it away, it attracts it, as if to hold it back.

This is a consequence of the conservation of energy: induction does not create free energy. To produce the current, work must be supplied — the work of fighting against this opposition. Current is harvested only by pushing against Lenz's law.