How does current flow in a circuit with a capacitor? When a capacitor is connected to a battery, current starts flowing in a circuit which charges the capacitor until the voltage between plates becomes equal to the voltage of the battery.
Since between plates of a capacitor there is an insulator/dielectric, how is it possible that current flows in a circuit with a capacitor since according to Ohm's law, current is inversely proportional to resistance and an insulator by definition has a big resistance, so we basically have an open circuit?
 A: 
how is it possible that current flows in a circuit with capacitor
since according to Ohm's law current is inversely proportional to
resistance and insulator by definition has a big resistance, so we
basically have an open circuit?

The short answer is because electrons can flow to and from a capacitor without the electrons having to pass through the insulation between the plates. The following qualitative explanation is offered:
Assuming the capacitor is not initially charged, then before it is connected to the battery each metal plate has an equal amount of protons (positive charge) and highly mobile electrons (negative charge) so that each plate is electrically neutral and there is no voltage (potential difference) between the plates.
When the capacitor is connected to a battery, the positive terminal of the battery attracts electrons off of the plate connected to it moving them to the positive terminal of the battery. This leaves a deficit of electrons on that plate making it positively charged.
Simultaneously, the negative terminal of the battery supplies an equal amount of electrons to the plate connected to it giving it a surplus of electrons making the plate negatively charged.
This moving of electrons from one plate to the positive terminal battery and from the negative terminal of the battery to the other plate is the capacitor current. Note that the electrons do not travel through the insulating material (dielectric) between the plates.
You can think of it roughly in terms of the electrons being "pulled" off one plate and "pushed" on to the other by the force of the electric field produced by the battery, but that the charges get "stuck" on the plates because they can't get past the insulating dielectric.
Eventually, as you already appear to know, the battery stops moving electrons between the plates when the potential difference across the plates equals that of the battery.
Hope this helps.
A: The removal of electrons from the capacitor plate connected to the + terminal constitutes a current. As those electrons are removed for that plate, there is an accumulation of electrons on the other plate. That movement of electrons constitutes a current.
The current stops when the potentials of the capacitor plates are equal to the potentials of the respective battery terminals. This does not happen instantaneously, but rather has a time behavior because the transportation of electrons out of and into the capacitor requires time, and the potentials depend on the charge imbalance of the plates.
A: The presence of a parallel-plate capacitor means that in part of the
circuit (only a small part; capacitors rarely have a gap as large
as one millimeter) there  is no movement  of electrons, only a buildup
of field (accompanied by electrons  if the capacitor is not a vacuum
type).   This is problematic, because there is a simple way of detecting
current, which is to observe the magnetic field that current creates,
and PART of the circuit now no longer has current.
The fact  is, that 'correction' to the magnetic field does not
exist.   The relevant Maxwell equation for current creating magnetism
has a term added to the current displacement current, which is the rate of change of the
electric field (like, the field inside the dielectric of a capacitor).
That addition to the equation is not just necessary for circuits,
it has the added side-effect that a changing electric field creates a magnetic field, even with NO charged particles in motion.
That term in the equation is why electromagnetic waves (light)
travels in a vacuum.   And, why charging of a capacitor is (in our
measurements) indistinguishable from continuous flow of current in
a circuit.
Literally, we can see the sun shine, because a capacitor gap in a circuit
isn't distinguishable from continuous current through a circuit.
A: Since this is a physics q and a, a physics explanation is in order.
There are two kinds of current.
Conduction current is a net flow of charges. It is was people usually think of when the word "current" is used
Displacement current is another form of current,  first recognized by Maxwell. Displacement current plays an essential role in Maxwell's equations. Displacement current density is proportional to the time derivative of the change of electric flux density.
When electron current flows into one side of a capacitor, the electrons accumulate, as there is no place for them to go. As the electrons accumulate, the electric flux density changes. This causes, or perhaps "is" a displacement current.
On the opposite plate of the capacitor, a similar process occurs, but with opposite electrical polarity.
The displacement current flows from one plate to the other, through the dielectric whenever current flows into or out of the capacitor plates and has the exact same magnitude as the current flowing through the capacitor's terminals.
One might guess that this displacement current has no real effects other than to "conserve" current. However, displacement current creates magnetic fields just as conduction current does.
This answer is perhaps more than one might want to know, but it is part of the story of electricity that is worth telling.
A: A capacitor does indeed block direct current (DC). However appreciable alternating current (AC) can flow when the period of oscillation is less than the charging time of the capacitor.
A: Pumping electrons into one plate of a capacitor causes loose electrons on the other plate to be repelled when they "see" the other electrons coming in. This causes a brief pulse of electrons to flow out of one plate when electrons flow into and populate the other plate. For large plates, this brief pulse is long, and for small plates the brief pulse is short.
This means brief pulses of AC current can easily flow through a capacitor, while steady-state DC current is completely blocked.
A: A capacity (condenser) can be charged initially during the transitory build up of the charge on the capacitor when closing the circuit. Typically:
$ = .$
With:
$R$: Resistance of the circuit.
$C$: Value of the capacity.
In fact, it also happens to be the mechanism behind the ability of the capacity to let through the time-variable component of a signal (current induced by the variation of a charge on the sides of a capacity), while blocking the constant component thereof.
