See Apple's http://support.apple.com/kb/HT2090.
If your problem continues after considering all relevant suggestions there, use your Mac's Disk Utility to repair permissions. Then download and apply the latest Combo Update for your Mac OS. the current ML Combo is at http://support.apple.com/kb/DL1659.
Repair permissions again immediately following the system restart that completes the Combo update.
Properly applying the Combo version of updates can sometimes correct software problems, but, if your trouble is caused by a hardware problem, there are no user serviceable parts of the camera. For hardware help, contact Apple or an Apple-Authorized Service Provider.
(Over time, Apple has changed the built-in camera's name on newer Macs from "iSight" to "FaceTime" and then to "FaceTime HD." Regardless of the name of your Mac's built-in camera, the same info and troubleshooting applies.)
Apple has been aware f this problem and just doesn't care. Their fix is to contact the developer.
See KB TS4552, http://support.apple.com/kb/TS4552
If you need to chat with someone on a PC, you need to purchase another laptop.
MacBook Air (Mid 2013): FaceTime HD Camera may not work with some applications
Some applications not developed by Apple may not recognize the built-in FaceTime HD Camera or may display a black screen on MacBook Air (Mid 2013) computers. This may also occur with Final Cut Pro X v10.0.8.
Update any affected application to the latest version. Contact the developer for assistance with applications not developed by Apple.
As a work around for Final Cut Pro X or other OS X video applications, you can use QuickTime Player's New Movie Recording menu item. Use this to capture video from the built-in camera and then import the clip to the application.
I just noticed from Skype.
There is an issue with Skype not recognizing the MacBook Air camera under OS X 10.8.5. Apple and Microsoft (Skype's owner) are aware of the problem. Here's some documentation:
Note that other apps such as FaceTime and Photo Booth continue to work just fine with the MacBook Air camera under OS X 10.8.5.
i have a late 2010 MBA. neither skype nor photobooth recognizes the camera. i tried updating to mavericks but it still doesnt work....
Upgrading your Mac OS to 10.9 wouldn't help the problem you describe. Instead, start with How to Troubleshoot iSight. (Over time, Apple has changed the built-in camera's name on newer Macs from "iSight" to "FaceTime" and then to "FaceTime HD." Regardless of the name of your Mac's built-in camera, the same info and troubleshooting applies.)
A more likely fix than changing your Mac's OS would be to install (or reapply) the latest compatible Skype update. If updating the third-party Skype app did not resolve your trouble, you could then have restarted your Mac and reinstalled the latest version of your Mac's OS. Having already upgraded to OS X 10.9, that mans you should Reinstall OS X Mavericks if updating Skype does not fix your trouble. FYI, I always use Mac's Disk Utility to repair permissions before and immediately after installing any OS update or upgrade.