The recent Windows 10 Anniversary Update had the unintended consequence of disabling some people's webcams in Skype and other video chat apps.
Reinstalling the camera's drivers or checking for any available updates is the best way to solve this problem. To do so, open Device Manager (search for it in the Start menu) and select the 'Imaging Devices' section. Right-click on the camera you're using, go to Drivers and look for manufacturer updates.
Microsoft is aware of the issue and is working on a solution that will be patched into Windows 10 directly. This solution was supposed to be released in September, but that didn't happen – hopefully, the patch will be available soon.