As capacitors charge, the negative box keeps filling with electrons while the other (positive) box loses any electrons still in them due to repulsion from this negative box. Since the negative box is relatively empty to start with, electrons fill in very quickly.
The negative plate repels electrons, which are attracted to the positive plate through the wire until the positive and negative charges are neutralized. Then there is no net charge. The capacitor is completely discharged, the voltage across it equals zero, and there is no discharge current. Now the capacitor is in the same uncharged condition.
There is no potential difference from each plate to its battery terminal, however, which is why the capacitor stops charging. The negative and positive charges on opposite plates have an associated electric field through the dielectric, as shown by the dotted lines.
In the figure below, the capacitor is neutral with no charge because it has not been connected to any source of applied voltage and there is no electrostatic field in the dielectric. Closing the switch, however, allows the negative battery terminal to repel free electrons in the conductor to plate A.
When a voltage is applied on a capacitor it puts a charge in the capacitor. This charge gets accumulated between the metal plates of the capacitor. The accumulation of charge results in a buildup of potential difference across the capacitor plates. So there is a voltage built across the capacitor.
A capacitor can store the amount of charge necessary to provide a potential difference equal to the charging voltage. If 100 V were applied, the capacitor would charge to 100 V. The capacitor charges to the applied voltage because it takes on more charge when the capacitor voltage is less.