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 bulbs.
If you remove one of the light bulbs from the light bulb holder, although the holder remains attached to the cables the circuit "opens". If you look at the holder carefully, you will see that it 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 opens.
You will have the same effect if you disconnect one of the cables.
The same happens when one of the light bulbs is burned out. The circuit opens because the wire inside the bulb is cut and thus the circuit is interrupted. The same will happen if you take a pair of scissors and cut one of the wires.