A diaphragm pump (also referred to as a Membrane pump) is a positive displacement pump that runs on the mixture of the reciprocating actions of a rubber, thermoplastic or teflon diaphragm and suitable valves on either side of the diaphragm
A double diaphragm is a positive displacement pump which utilises two flexible diaphragms that reciprocate backwards and forwards, creating a temporary chamber, which both draws in and expels fluid through the pump. 
… Both diaphragms that are linked by a shaft through the center section where in fact the air valve is located.
Though it depends upon the pump and application, a positive displacement pump, is generally a self-priming pump. … For Air Diaphragm Pump example, air managed diaphragm pumps self-primary by creating a pressure differential in the diaphragm chamber. This draws in atmosphere and pulls the liquid in to the suction port.