Basic Concept: The Plate Capacitor

Two metal plates equal in size stand opposite each other a certain distance apart, separated by air or some other insulator. Voltage is applied:

The capacitor takes up both negative and, on the other hand, positive charged carriers on the surface of it's plates. It stores electric charges, so to speak.
(Mechanical analogy: storage receptacle for fluid).

Direct current is normally unable to pass through this structure; due to the insulator (dielectric) lying in between, the electrons cannot get from one plate to the other. This is only possible in the case of very high voltage, and then a breakdown occurs due to ionization.

Note: The capacitor is usually infinitely resistant to direct current (DC).

On the other hand, the capacitor lets alternating current (AC) through.
The current cannot flow directly through the dielectric even here, but because of the alternating charging and discharging of the plates, charged carriers appear to be transported through the capacitor.
Of course, the capacitor presents some resistance to alternating current too, this is dependent on the frequency.

Note: At lower frequencies the capacitive resistance is higher. At higher frequencies the capacitive resistance is lower.