In reverse order... can a bad TC render a transmission inoperable? Yes it can. You usually have SOME operational capabilities with a bad convertor, depending on why it's bad.
How they work. Well, remember when the engine is running it is turning the flywheel and so outputting rotational energy. When you are sitting stopped at a stoplight, you are not using any of that energy. So we have to somehow alter any direct connection between the engine and the wheels. In a manual transmission car you put in the clutch and accomplish that. In an automatic the TC accomplishes that for you.
How? Well let me use an analogy. Think of a housefan. No think of 2. Start one of them... think of that as the engine - outputting rotational energy. Now put the other one near the first one.... in line... so the breeze produced hits the 2nd one. What happens? The blade on #2 starts turning. At low speeds on the #1 fan you can easily stop the dead (#2) fan's blades using your fingers. (Like the brake at the stoplight stops the car from moving). Speed up the powered fan and the other fan speeds up too... in fact, at some point in the speed up process of fan #1 you ain't gonna want to put your fingers in that 2nd fan to try and stop it! THAT is a very basic idea of a TC. It also multiplies torque but that is another matter. The guts have a turbine in it, very much like a jet engine! Your "fan". One turbine is attached to the shell and hence flywheel, the other is attached to the transmission. Voila, your two fans.
A TC uses fluid instead of air which causes a strong viscious coupling way better than air, and at around 40 mph both "fans" are turning at close to the same speed. That is when modern TC's add a twist - a lockup clutch. Once that is engaged the two fans are for all intents welded together as one. Which is why if your TC lockup does not disengage it stalls the engine... it is like leaving the clutch in on a manual as you come to a stop.
That is a pretty basic intro. A good website description can be found here:
http://auto.howstuffworks.com/torque-converter1.htm
Enjoy!
ps. One other point. All that fluid moving around in the turbines of a TC has a lot of fluid friction and hence produces a lot of heat! Lockup helps stop the heat production in a TC. This is also why some computer controlled trannies often "inhibit" lockup until the transmission is warm... they "leave the heat on".