YouTube updated its data API on April 20, 2015 affecting devices manufactured in 2012. This means the YouTube app will no longer work with older devices like the first- and second-generation Apple TV. Other devices affected include iOS devices like the iPad and iPhone running iOS 6 or earlier and Sony TVs and Blu-ray players.
You may have to update your software if the YouTube app is not working on your third-generation Apple TV. To do this navigate to Settings > General > Update Software on your Apple TV and click "Download and Install".
Owners with older iOS devices can still access the YouTube website by using the mobile browser on their device.
You can see a list of other affected devices and solutions here.