As far as using something that taps into the stock oxygen sensor as a part-throttle tuning tool, inspyral is right. The ECU can run in either "closed-loop" or "open-loop" mode. In closed-loop the ECU picks how much fuel to use pased on RPM and MAP sensor voltage, that's it. At WOT it switches over to open-loop and uses O2 sensor data to compensate for any difference from the intended ratio. Since the ECU is only in open-loop mode at WOT, the sensor is only really designed to give an accurate reading at WOT. A "wide band" sensor simply means that it gives an accurate reading at more than just WOT.

Stoichiometric ratio is when you have exactly enough air to combust all of the fuel. For gasoline this is 14.6:1 (air:fuel). Rich means you have more fuel than that, lean means you have less fuel. Running lean (higher than 14.6:1) has the advantage of more fuel economy and at the same time produces more energy in the form of heat. More energy = more horsepower. If you run rich (lower than 14.6:1) not all of the fuel is combusted. If you stand behind an old carbeurated car and can smell gasoline in the exhaust that's because it's running rich. You don't really gain any power from running rich because no additional fuel is being combusted. The extra fuel helps to lower cylinder temps which allows more aggressive tuning of things like compression ratio and ignition timing.