Outboard water pumps are located on top of the lower unit at the base of the driveshaft. The driveshaft turns a flexible neoprene impeller inside a specially-shaped housing to pump the water. The impeller is locked to turn with the driveshaft using a woodruff key or wedge. As the engine RPM increases, the water-pump pumps more water at a higher pressure. A small portion of the water pumped comes out as a small stream through the telltale at the side of the powerhead to let you know the pump is working.
This video shows a method of testing if the water pump is working.