Your TT has two systems. 120 volt and 12 volt.
The lights, water pump, radio operate on 12 volt.
The refrigerator and furnace operate on propane AND 12 volt (or the frig can also run on 120 volt).
When you are plugged into shore power (120 volt), your converter keeps your battery charged and also runs your 12 volt systems AT 12 volts. It converts 120 to 12 volt.
Your electrical outlets, air conditioner and microwave all run ONLY on 120 volts.
Your water heater can run either on propane or 120 volts.
An INVERTER will convert 12 volt DC to 120 volt AC and it can then run things like your TV, coffee pot, DVD player, etc. However, it cannot run your Air Conditioner, and it can only run your microwave if you have a very large battery bank and a very high wattage inverter.