A few months ago I was in your position... had the computer all done, but when it comes to power in the car, I was clueless.
Not really a dumb question, but seraching for 'inverter' would have shown a boatload topics.
Anyway, you need to figure out first how to correctly wire the inverter to the battery ... I have a Statpower 300 that has a great diagram and pictures on how to do it. (I haven't done that yet as I just ordered a shutdown controller from Maestro and will do all the wiring at once)
This brings us to the Shutdown Controller. (Use the search feature to learn more about it) ... It is attached between the inverter and car battery. It sees the ignition to the car is cut. It will keep the inverter powered and send a shutdown command to the PC. You know it takes 10 seconds to shut the PC down, so a timer on the SC is set to 10 seconds. After 10 seconds, the power to the inverter is cut and everything is off.
I could be wrond about a few of the SC details, but I haven't received mine yet, just going from what I read on postings.
What kind of power supply you got running? You need nothing more than 150 watts. Even a standard 150W PS is less than $10. Use the instead of a 250W or 300W PS that's standard in most PCs.
Search the forums for words like 'inverter' 'power supply' 'shutdown controller' and I assure you all questions will be answered.
Hope this helps,