First approximation, others will be along to refine, add, change and correct.
The Power Commander's (PC) primary job in the FJR is to alter fuel injection mapping based on % throttle opening and RPM. The PC adjusts the fuel injection pulse +/- some % from the stock ECU map. It has an optional 'accelerator pump' feature that enriches FI upon pinning the throttle. There are many basic FI maps available for the PC. The real power of the PC is when you put Exhaust Gas Analyzer (EGA) bungs in each head pipe then run your FJR on a chassis dynamometer and have it create a custom map that matches your engine exactly.
The various maps add or subtract fuel to compensate for the stock ECU map that is pretty much dedicated to emissions first. Some maps alter the ECU map to enhance throttle smoothness. Some maps can be set to optimize either economy or power. If you make changes to your FJR such as pipes and intake filters the PC will let you match the FI to the new engine flow rates. Someone will be along with a link to Wally's Smoothness map, I'm sure!
The O2 sensor measures the amount of oxygen in the exhaust. The O2 sensor only works once warmed up. It supplies the ECU with a voltage that is proportional to the amount of oxygen so that the ECU can adjust FI to meet some value that equals optimum emissions. In the FJR, the O2 sensor is only read by the ECU when the engine is fully warmed up and running at a basically steady rate. The O2 sensor gets disconnected to prevent the ECU from compensating for FI mapping changes made by the PC.
The Barbarian Jumper (BJ) makes a minor adjustment to the FI system, the adjustment result is most predominant at or near idle. The BJ offset to the basic FI map is a one dimensional adjustment. The PC is more of a '3D' adjustment because it takes in consideration all RPMs, throttle position and rate of throttle changes. It would probably be best to return and BJ adjustments back to stock before installing a PC but in the end, it may not matter much.
The PC software lets you read all the maps, and should you be an intrepid twiddler, it will let you change anything and everything. There are also quick adjust buttons on the PC itself.
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That should be another thread. Probably never ending and never resolved.