An open circuit means that the circuit is not complete. It is broken somewhere. It can be a gap remaining in the connection.
Suppose you connect two copper wires to a small bulb and connect the other end of one wire with a battery. You still have to connect the other end of the second wire to the battery to make the bulb glow. When this type of gap remains in a circuit, the circuit is called an open circuit.
As the connection is not complete, potential difference is not created between the two ends of bulb. So, electrons do not move in a specific direction and so the current does not flow.