Most YouTube content requires the Flash plugin... try troubleshooting.
Open System Preferences > Flash Player then select the Storage tab. Click: Delete All
Now uninstall the Flash plugin then reinstall new > Troubleshoot Flash Player | Mac OS
Launch Safari From your Safari menu bar click Safari > Empty Cache
Now try a video.
BTW... if you have the ClickToFlash extension installed, that can prevent Flash based video from streaming. It can also be installed as a plugin in /Library/Internet-Plug-Ins.
And check to see if Safari is running in 32 bit mode. Right or control click the Safari icon in your Applications folder then click Get Info. If the box next to: Open in 32 bit mode is selected, deselect, quit then relaunch Safari.