Oil pumps may be driven from the camshaft or the crankshaft.
In a rotor-type oil pump, an inner rotor drives an outer one. As they turn, the volume between them increases. This larger volume lowers the pressure at the pump inlet.
Outside atmospheric pressure is then higher. This forces oil into the pump, and it fills the spaces between the rotor lobes.
As the lobes of the inner rotor move into the spaces in the outer rotor, oil is squeezed out through the outlet.
The crescent pump uses a similar principle. It is mounted on the front of the cylinder block.
The inner gear is on the end of the crankshaft which then drives the pump directly. An external toothed gear meshes with this inner one. Some gear teeth are meshed but others are separated by the crescent-shaped part of the pump housing.
The increasing volume between gear teeth causes pressure to fall. Oil is then taken through the intake port, and carried around between the gears and crescent, then discharged to the outlet port.
Similarly in a geared oil pump, the driving gear meshes with a second gear. As both gears turn, their teeth separate, creating a low pressure area. Higher atmospheric pressure outside forces oil up into the inlet. The spaces between the teeth fill with oil. The gears rotate, and carry oil around the chamber. The teeth mesh again, and oil is forced from the outlet toward the oil filter.