After the original (flawed) update to OS X 10.8.5 was installed, the internal camera of a 2013 MacBook Air would not work in Skype and some other third-party applications. The flaw was fixed in a supplemental update. However, if you followed some questionable advice that was circulating on this site to hack your system before installing the supplemental update, then it will not restore camera functionality; in fact, the camera won't work at all. The solution in that case, most likely, is to install the 10.8.5 Combo update.
In the future, I suggest you ignore advice to replace, remove, or change built-in system components.
If you don't already have a current backup, back up all data, then reinstall the OS. You don't need to erase the startup volume, and you won't need your backup unless something goes wrong. If your Mac was upgraded from an older version of OS X, you’ll need the Apple ID and password you used to upgrade.
There are ways to back up a computer that isn't fully functional. Ask if you need guidance.
If you installed the Java runtime distributed by Apple and still need it, you'll have to reinstall it.
I experienced this same problem back in late 2013. I updated my software, and the problem was solved. I am now experiencing the same exact issue when trying to video chat on google hangout... Frustrating!