I am not sure what the wire is called, but the one that burned is located on the top of the alternator.
The fire? Really wasn't a big deal. I was driving the car to work and smelled hot electrical which was immediately followed by smoke in the interior. I pulled over and popped the hood. The connector and the insides of the alternator were on fire. So, I used a coat out of my car and smothered it out because I had nothing else. Luckily I wasn't far from home.
My guess is that the alternator shorted out or something and the wires got so hot that that crappy material on the connector just caught on fire.