To be precise, some accessories run on battery voltage usually 12 volts . The power from those comes from the batteries or a gen set. Shore power may be used to power a battery charger which will recharge batteries to a point. Although most ACs in this class are AC powered, there are dc only powered versions.
If your air conditioner is ac powered, it will work at the dock on shore power but not at sea unless you have sufficient battery capacity and a correctly sized inverter.