Your inverter is powered by your "house" batteries. It converts DC power to AC. The inverter is used when you aren't plugged into shore power and don't want to run a generator. Length of usage depends on your batteries and their condition.
Normally, an inverter powers only a select few of the coach AC receptacles. For example, mine powers only the entertainment systems and one additional outlet in the bedroom for an alarm clock, DVD or game console and not the refer or microwave.
I suspect in your 04 Dolphin that the inverter powers only the tv/vcr/dvd electronics. If so, leave it on all the time because it is probably the only power source for those things. If you have a larger model of inverter that powers other appliance and outlets, you can leave it OFF whenever you are running on shore power or genrator, and turn it on whenever you need 120vac power and do not have any other source.
There are two types of inverters for this discussion (There are two types for another one too).
In-line and stand alone.
Now, if it is a pass through then with the inverter switch on or off (Depending on the model) Everything works on shore power or generator power...
With the switch ON selected loads work on Battery when no shore or generator power is present.
This type of inverter should go into "Standby" when shore/gnerator power is present, and in some cases it also charges the battery (Kind of like a UPS on your computer)
Leave the switch on
The stand alone types have 12 volt in and 120vac out leads/outlets. They do not have the ability to sense shore/generator power and thus will keep on sucking power when shore/generator power is present.. On this type you turn it OFF if you don't need it.
The other two types are MSW and TSW, and that's a different thread.