Double acting means that it will compress the gas in both directions of the cycle, these boosters will have a piston on both sides of the barrel, single stage boosters will have 2 compression pistons that are the same size and volume and work in parallel. The advantage here is that the output volume is doubled. A two stage unit will have the second piston that is smaller in diameter, the gas is first pumped in the larger cylinder, compressed and then sent to the second cylinder where it is compressed even further thus working in series, the advantage here is higher pressures.
The large piston in the barrel uses LP gas to compress the high pressure gas by the machanical advantage of the larger surface area of the piston. When it reaches maximum stroke the piston makes contact with a pin which activates the cycling valve and dumps the LP gas from the barrel and then the piston is forced back back to the starting position where it makes contact with another pin and the process is repeated. The pilot valve is used to control maximum output pressure.
ZDD