Gadling

The APU is a small turbine engine located in the tail of the aircraft. It can power the aircraft's electrical systems when the main engines are shut down, and provide starting power for the main engines. Being smaller than the main engines, it burns considerably less fuel.

When stopped at a gate, an aircraft's systems can use the airport's electrical power through an umbilical cable plugged into the bottom of the aircraft, but a high pressure air feed is needed to start the engines if the APU has been shut down.

I feel like pilots often say the APU on the Super80 isn't working, or that it's not working well? Would a plane stay in service if the APU was not working?

The APU, while convenient, is not critical to aircraft operation.

Without the APU (either if it is inoperative or just off), a ground "airstart cart" (or other high pressure air source) must be used to start the main engines; an aircraft cannot "turn itself over" without the APU. Generally, at the beginning of the day, an aircraft is "cold" with no engines active and an airstart cart is used to start the engines and fire up the APU.

Later in the day, when on the ground the main engines can be shut down when practical in order to save fuel; if the APU is operative and running the main engines can be restarted at the pilot's command, on a moment's notice. Without the APU, the main engines have to be started with the help of ground crew and an airstart cart.

The simple solution to this is to keep at least one engine running in order to maintain hydraulic pressure and power the aircraft's electrical systems. With one main engine turning, bleed air can be fed into the remaining engine(s) to power them up. This uses more fuel than powering down the main engines and using the APU "on the ground", but is not at all dangerous and does not impact aircraft performance.