A pressure pump is a pump with some sort of electrical switch on it which turns the pump off when the system pressure reaches a pre-set point, sometimes this is fixed and sometimes it is adjustable. When your pressure pump is running and a tap on the system is open, the water flows out of the tap.
When the tap shuts, then the water stops flowing and the pressure in the system builds. When the pressure reaches the “off” setting of the switch or controller, then this turns the pump off and the pipes remain pressurised. As water is not very compressible and pipes only store a small amount of water, when you open the tap again the pressure drops very quickly and only a small amount of water comes out. When the pressure drops to the “on” setting of the switch, then the pump turns on again and starts pumping.