What is this green on and what does it do?

You're right, Your boyfriend is wrong. How unusual. The overdrive lets the engine run slower and more relaxed at higher speeds. This uses less fuel, and extends engine life. There is no reason to turn it off for normal driving. If you are in hilly areas, or find you want more acceleration for some reason, you can turn it off. Again, 99.9% of the time, you can leave it on.