The contacts are the little round pieces of metal on the end of the arm that moves, and the matching set on a fixed support.
Rather sounds like you have a bad pressure switch. If you can push the contacts together and the pump works properly, that is a problem. Just use an insulator to push on the arm that has the movable contacts. Don't let it jitter on and off. Bad for the pump.
If that works you want to relieve pressure in the system and remove the switch. Turn off power at the breaker. Remember the dieing part. Best to check with a meter to be sure that the breaker you turn off is the right one. Disconnect the wires. Remember where things went. You unscrew the switch assembly from the pipe it is attached to. Check to see if there is some sort of crap in there. If there is a chunk of something, you may get it to work by cleaning out the inside of the switch. The switches are relatively inexpensive.
To replace the switch, take it to a plumbing supply or tell them the make and model. The two important parameters are the on/off pressure and the type of fitting that it is mounted on. There are also normally open and normally closed switches but yours is most likely one that is open when not trying to run the pump. A big box store may have one.
I do not understand why the pressure would not go to zero when you leave a faucet on. There may also be something else going on between the tank and the rest of the house. Or the pressure Gage is wrong. If you let water run until water stops running, the tank should be at zero pressure. Could be clogged pipes from hard water; rusting galvanized pipe, filter, water softener, water devils cavorting in your pipes, etc.