How does current flow through capacitor?
There is often a curiosity and misconception about capacitor. A capacitor has an insulation material between electrode plates, and an insulator has a property to block electric current. Then how does a current flow though a capacitor? The question is natural, since we always talk about capacitor current and it can be measured on an instrument. The mechanism of current flow is different from that through a conductor, or through a continuous current path.
To understand capacitor mechanism, let us consider construction and working of an ideal capacitor.

The dielectric of a capacitor placed between electrode plates is essentially an insulation medium, with an additional property of polarization. The atoms or molecules of dielectric material are formed of a nucleus and an electron cloud around it. These two have common center point, and as such the whole molecule is in equilibrium around this center.
When a voltage is applied to capacitor terminals, an electric field is created in the dielectric medium, directed from positive plate to negative plate. The electron cloud gets attracted towards positive plate (opposite polarity), and the nucleus gets attracted towards negative plate. The whole length and body of dielectric gets polarized in this manner. Â To balance the negative charges (electrons) appearing at positive terminals, positive charges get accumulated on positive plate and the charges are held in equilibrium. Similarly, at negative terminals, electrons get accumulated on plate.

Thus, when a voltage is applied across capacitor plates to increase its voltage, some positive charge is accumulated on positive plate, and an equal and opposite positive charge is removed from negative plate (or electrons are added to negative plate). The process of addition of charges, drawn from supply source, continues till capacitor voltage reaches the applied voltage. The addition of charge and removal of equal charge from negative plate (movement of charges) is perceived as a current by an external observer.
The current stops when capacitor voltage reaches applied voltage. Thus no current is seen to flow once charge transfer stops. Hence capacitor is said to block DC steady current. The process of addition or reduction of charges is through orientation of dipoles in the dielectric, which always try to align with external applied field. Once equilibrium is reached, dipole movement stops, and plates have constant fixed amount of charge. The extent to which these two centers gets displaced from one another depends on the nature of material and ease of polarization. Ideal insulators do not undergo any polarization at all, but there is no ideal insulator in practice. The materials considered as dielectrics have significant polarization, which make them suitable to make a practical capacitor.
An AC current poses a different situation. The voltage across plates is constantly changing. So the charges on plates are constantly trying to adjust to external voltage by accepting or returning the charges (since the dipoles are constantly changing their orientation). External voltage keeps changing every moment, and dipoles in the dielectric therefore are constantly in the process of adjusting to this ever changing field. This means the charges on electrode plates also keep changing, following the voltage curve with time. This continuous change is reflected in external circuit as continuous movement of charges, meaning a sustained AC current. This current can be measured by any instrument.
Looking at these phenomenon, it becomes clear that in reality, there is no physical movement of charges from positive plate to negative, or vice versa. In the space between electrode plated, only the dipoles in dielectric material keep oscillating between two extreme positions of alignment in electric field.
So for all practical purpose, all movements of charges are external to capacitor, and no current physically flows through a capacitor. What actually happens in a capacitor with AC voltage is continuous change in orientation of electric dipoles in the dielectric, with corresponding change in charges on plates.
This way a capacitor is seen to block a steady state DC current, and carry a steady state AC current.
We have considered an ideal capacitor here, whereas a practical capacitor has some resistance, of a very high order (insulation resistance), between pates. It also has resistance of connecting wires, resistance of metal conductor carrying the current to terminals, resistance of joints / connections etc. This causes a resistive current, though too small, through the dielectric. This causes some power loss, reflected in loss factor of a capacitor.
The nature of dielectric, and electrode material, decide the magnitude of losses and hence loss factor of capacitor. Therefore, the losses differ with the type of capacitor.