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Deadband is also known as hysteresis and is the amount of a measured variable (pressure, temperature, etc.) between the point where a switch closes and then re-opens. It can be implemented is software if you have an analog input (i.e., open at 1.02 PSI, close at 0.98 PSI) or designed into a switching mechanism. Often the deadband of a switch is fixed and cannot be adjusted. A typical example is in a wall thermostat in your house.
The enclosure containing the micro-switch and other accessories of the pressure switch is called the switch case or housing. For a pressure switch giving out an electrical output one or more tapped connections are provided to allow the cable to be brought into the housing via a suitable gland.