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Usually a capacitor is made of two conductors/plates having $+Q$ and $-Q$ charges My book says that
$$\text{capacitance} = \text{charge on either conductor} / \text{potential difference between the conductors} \, .$$
If there is some potential difference between the plates, then charges will flow from one plate to another, then how can there be same charges on both the plates of the capacitor?
Also do charges move through a capacitor when connected to a circuit?
First, it is not the "same charge". If you had the same charge on the two identical plates you would not have any potential difference between the plates. The potential difference is due to the difference in the signs of the charges (one positive and the other negative).
The reason the charges do not move from one plate to the other is that the plates are separated by an insulator. If you connect the two plates with a metallic wire to create a conductive path the charge will flow and you may see a spark. But for the charge to move through an insulator (or dielectric) you need a quite high electric field (see dielectric breakdown field). If you look at the label of a capacitor, you will see the maximum voltage indicated there. As long as the voltage is lover than this nominal value, the charge cannot "break a path" through the insulator and the capacitor stays charged, as designed. But if you try to charge to 100 V a capacitor rated for 10 V is very likely that the charges will move from one plate to the other through the insulator.
In addition, when it breaks down the insulator (if it's a solid) may change its nature, chemically, like burning. So a permanent conductive path may appear. If the insulator is a gas, you can have a gas discharge like in lightening.
The path between two plates is completely insulated hence the charge do not move between plates. One of the plate can be considered to be connected with the ground i.e. charge reservoir. If you connect one plate to a power source it will induce some charge on the connected plate and the equal amount of the opposite charge is then induced on the other plate. The induced charge on the grounded plate is supplied by the ground.
The amount of charge accumulated on the plate on the application of voltage is proportional to the voltage i.e.
$Q\propto V$
now that constant of proportionality i.e. the ability to store the charge on the application of voltage is known as the capacitance. It can be seen that the charge accumulated on the simple plate capacitors is proportional to the area of the plates and inversely proportional to the distance between the plates (if the distance is small compared to the size of the plates). It can also be seen that the capacitance can be increased if you place some high electric permeability material between the plates.
If you apply continuously varying voltage on the plates then the induced charge will also very and which will result in the current flow in the other arm and that is the reason behind the current flow through the capacitor. Capacitor blocks the DC current and allow the AC.
Well, it could go either of two ways. Directly through the capacitor, or around, through the rest of the circuit.
It can't go across the capacitor because there's a dielectric there. (Air, maybe.)
Actually, you're right---it will discharge through the rest of the circuit. That could be a resistor or inductor, for example.
The exception is when you first charge the inductor with a battery. In that case, the potential of the battery cancels the charged potential of the capacitor.
