Inductors are energy storage elements like capacitors, except that they induce voltage. (The purple lines in the diagram to the side are magnetic field lines produced by the inductor at a single moment in time)

With the change of current through the coiled wire of an inductor, a magnetic field is produced, one which creates a surrounding voltage. In the presence of a DC current, the inductor acts just like a short circuit. But in an AC circuit, the inductor induces a voltage that is proportional to the derivative of the changing current.

This property is thanks to Faraday's law, one of Maxwell's equations (Originally discovered/described by Faraday, and then collected and unified by Maxwell, hence the two names). One way of stating Farady's law is :

"The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit" (thanks Wikipedia).

The changing magnetic field produces a curling electric field; since we curled a wire (the inductor itself) around the magnetic field running through the center of the inductor, this curling electric field induces a voltage in the circuit.