Ran into an issue this evening trying to use my FaceTime camera via my Thunderbolt Display. After launching the video conference software and connecting to the call, camera would not share any video. It simply showed a black screen.
Some quick Googling revealed the answer. Run this command in your terminal:
$ sudo killall VDCAssistant
Then restart your video conference software (FaceTime, Skype, whatever). Your camera should begin sharing video right away.