Think the capacitor as a black box you see from outside. If you make any change in the applied voltage across this black box, you see electrons going into the device from one side, and electrons (the same amount) leaving the device from the other, so, you describe it as a current flowing through the device.
Now, you open de black box and find the capacitor, who clearly "insulates" the two terminals. This strikes you at first, because your description involved a current flowing through... what? air? This apparent contradiction is solved by remembering that the capacitor "stores charge" (actually it stores energy by separating charges... net charge always remains zero!). The "illusion" of a current going through the device rests on the fact that the electrons going into it are actually being stored on one plate (but it could be any shape) and electrons leaving it were "idling" in the other plate and now are leaving space for "positive charges" (lack of electrons, in this case).
(yungman points out that this current is called "displacement current", but I do not agree. "Displacement current" is a not-so-fortunate name given to the derivative of the D field, which in simple cases is proportional to the Electric field. I think the current we are talking about here is the ordinary one.)
Interesting conclusion: the "insulation" that a capacitor provides is only for constant voltages (usually called DC). Constant voltage ---> no current flows (you actually have to waits some time after the moment you apply it to let the charges accommodate). With varying voltages, charges never stop accommodating and you always have some current. Indeed, for very rapid changing voltages, the capacitor is like a short-circuit.
Electrons flow in an inductor in the same way the do in resistors (indeed all resistors have some inductance and all inductors have some resistance*), with the addition that they "feel" an additional force counter-acting upon them. This force is the result of an "induced" electric field because of the varying magnetic field, which in turn is generated by the very flow of the electrons (who came first? chicken or egg?). Energy is stored in an inductor by setting a current through it. Faraday's Law tells us that any change you attempt in that current, will be opposed by the fields, so you need energy to do it. Once the current is set, you can cut it very fast and energy will be liberated in form of a spark maybe (if a few conditions are met).
I think yungman gave you a fair idea of why the voltage is "behind" current in a capacitor. Remember that voltage is proportional to charge (V = Q/C) and charge is the "accumulation of current", so you need some current to go first to build up some charge to sustain voltage.
An inductor has some "current inertia" because any attempt to change it is counteracted by an induced electric field. If you apply some voltage across an inductor you have to wait some time before charges react to it, i.e., to the opposed field to go to zero as the current is established.
The exact dynamic of this is hard to tell without equations.
* At sufficient high frequencies, everything behaves as having resistance, inductance and capacitance. This happens when the physical dimensions are comparable with the wavelength of the "voltage" wave. The very notion of voltage stop making sense and you have to put Kirchhoff laws aside and work with distributed models or directly with Maxwell Equations.
I don't know if this was the explanation you were looking for. Sorry if I told you things you already known.
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