Pressure Vessels or Pressure Tanks are mainly used in conjunction with pressure boosting sets. Their ability to hold water at pressure prevents the need for the pump to be switched on constantly during the day for quick, short periods at a time.
For example, if someone were to wash their hands, water would initially come from that stored inside the pressure tank, if the tap was then closed, the pressure vessel would automatically refill from the mains pressure without the need for the pump. The pump would only need to kick in once the pressurised water from the tank has been used ie. if you were to fill a bath or use the shower.
Pressure vessels can also protect the pump by absorbing any shock to the system ie. if a tap is closed too quickly.