You can't have electrical current unless the circuit is closed. As soon as the circuit closes, the free electrons move from the point with low potential to the one with high potential that means from the negative pole of the battery though the cables and the light bulb to the positive pole. Therefore electrical current passes
through the light bulb.
If you unscrew one of the light bulbs, although the lamp base remains attached to the cables the circuit "opens". If you look the base carefully, you will notice that the base has two ends. The one touches the base of the lamp and the other the screw. When the light bulb is removed, there is no contact between the two ends of the base, therefore the circuit is open. This will not influence the operation of the other sub-circuits.