This is a tank that has a bladder type diaphragm inside which is used to keep pressure in your water system. As water is pumped into the tank, the diaphragm is pushed further up with greater and greater tension. After your pressure pump shuts off, the diaphragm keeps your water system pressurized.
As you use the water in your house, the diaphragm is under less and less tension, which means the pressure is going down. Once the pressure gets down to a certain PSI, the switch for the pressure pump turns the pump on and it builds up pressure again.
With any water system that uses a diaphragm tank, the larger the tank is, the better. Larger tanks enable you to go for a longer period of time in between the pump turning on. This is more efficient with electricity and can give more time to deal with mechanical problems that might arise. Tank sizes range on up to over 80 gallons (nominally).
For more details on how to choose the right tank for off grid use, visit Off Grid Water Pressure Tanks.