How do inductors produce current? This is how I understand batteries and capacitors. The terminals produce an electric field which causes the electrons to move in random directions. The surface charge builds up causing new electric fields to be created, which forces the electrons to move in the direction of the wire, creating current.
But with inductors, there is no negative or positive terminal (as in an end with a surplus or deficit of charge), so, on a microscopic level, how does it produce current?
 A: An inductor stores energy in a magnetic field.  After current has been flowing in the inductor for a period of time, it has built up a magnetic field around the wire making up the inductor.  In that state the inductor offers no opposition to current flow.
If it were then disconnected from it's energy source (battery perhaps) then the the magnetic field will start to shrink.  It's energy will go to move the electrons through the wire.  Thus, the inductor will produce a current for a period of time after it's disconnected.
An aside, the positive and negative terminals on some capacitors are for protecting capacitors due to how they were built (if you switch the terminals you might damage the capacitor) not what a capacitor theoretically is.  From an ideal standpoint, capacitors don't have polarity.  This is similar to inductors.
Another aside, electrons will move randomly because of the temperature and the material they are in (metal).  The random movement is not from any applied electrical field or whatever.  But, the electric field from something (battery perhaps) will cause the electrons to move in a uniform direction on average.
Now, at the moment I'm not able to give a blow by blow account of what's happening in an inductor from a particles perspective.  I think that would be a rather advanced kind of analysis.  But, rest assured, the relationship between the electric field and magnetic field that causes inductors to work is a very fundamental thing governed by the Maxwell equations.
