A: is the previously noted Resistrol - a device for varying the intensity of the lamp head by adjusting the voltage flowing to it (a fancy rheostat) so you can adjust exposure times (if you need to slow down a printing time for burning/dodging purposes). It is really only useful when working with black-and-white on a condenser head. As pictured, it is currently plugged into itself, so it is not doing anything.
When you wanted to use the resistrol, you'd plug it in to B, and then run the lamphouse through the resistrol. The dangling switch operates the current running through B, so it seems to do nothing because you're not using B.
C: is a transformer, but for what I don't know. I would have thought it was the voltage stabilizer for your cold light head, but it may have nothing to do with that.