Grounding is one of the most misunderstood concepts of electrical wiring, even among professional electricians.
On the DC side, grounding refers to running a wire from the negative battery terminal to the chassis of the vehicle. On cars, vans, trucks, etc., this is done primarily as a cost saving measure, allowing the vehicle manufactures to run one wire for every device rather than two.
On the AC side, the ground wire serves two basic functions.
The first is to protect the electrical system from excess current induced into the wiring from a lightning event by providing this excess current a path to the Earth. This is why buildings have grounding rods pounded into the ground where power enters the building. Take note that I said a lightning event rather than a lightning strike. A lightning event is lightning strike in the general area, close enough to induce current into electrical systems. A lightning strike is a direct hit and will fry a building's electrical system regardless of whether there is a grounding rod present. Both of these are largely non-issues when dealing with an AC power system on a vehicle. The vehicle's tires insulate the vehicle from the ground, making a grounding rod unnecessary.
The second purpose of a ground wire is to provide a path for the AC power to flow back to it's source in order to trip the breaker or blow the fuse when something short circuits. In this case, the inverter you install on your vehicle is the source, so all your wiring needs to have three conductors. You Need a Hot, which carries electricity to your devices, appliances, etc. This wire is usually, but not always going to be black, red or blue. Next you need a Neutral which is a path back to the source. This wire should be either white or grey. Finally, you need a ground wire, which should be either green or bare copper. This wire is not needed for your electrical system to work, but is an important safety feature. It will provide a path for the electricity to get back to the source (inverter) to trip the breaker or blow the fuse in the case of a short circuit. It will also trip the breaker or blow the fuse if the Hot wire comes into contact with the metal housing of whatever device you have plugged in, possibly saving your life. Sure, it's annoying having the breaker trip or the fuse blow, but I'd much rather have that happen than get electrocuted because my toaster had a frayed wire making contact with the housing.
Now with all this said, you really should run a wire from the ground post on the inverter to something metal on your vehicle. Once you've made that connection, your vehicle's body, frame and chassis components will become a path for the electricity to trip a breaker or blow a fuse. Once again, this could keep you from getting electrocuted when you go to open your door after the exposed wire you didn't notice on the extension cord running out your window gets blown into a lug nut by the wind.
