I had this issue with Mountain Lion that was fixed by the supplementary update ( MacBook Air (Mid 2013): FaceTime HD Camera might not work after OS X v10.8.5 update)
All was then okay with the world until, I think, 10.9 when it stopped working again. Tried everything suggested everywhere to no avail.
I upgraded to Yosemite this afternoon - and now the camera turns on for about 30 seconds, then the light goes off and all camera applications (Photo Booth, Skype etc) all report that no camera is connected. I can confirm that the FaceTime camera *is* listed under System Information - > USB, which it wasn't under 10.9, so this is progress of sorts. I have tried resetting the SMC as suggested in multiple places.
This issue is extremely frustrating and I've put a lot of effort into trying to fix it. I use the camera a lot for video calls with work and I'm about to have to resort to using an external camera on a supposedly high spec machine. I see a lot of other people are also having this issue on this thread and elsewhere.