An RC circuit could operate in either charge mode or discharge mode. In charge mode, the capacitor is hooked up to a potential difference (battery). Electrons leave the negative terminal and onto one of the two plates, the other plate become positive. The charge slowly builds up in the capacitor. The potential difference across it increases. The current would continue to flow but slowly diminishing as the voltage across the capacitor reaches the emf of the battery. Eventually, there would be no potential difference across the resistor, and no current flow. At this point the capacitor is fully charged.
The resistance, R multiplied by the capacaitance C, yields a product called the time constant of the circuit. This number corresponds to the time it'd take for the capacitor to reach 0.63 of its full charge
The equation of a simple RC circuit is given by:
Vc= E(1 - e^(-t/RC)
Where Vc is the voltage across the capacitance. E is the emf of the battery or voltage source, and t is the time