Do you get a plugin has crashed popup message when you go the youtube?
These things are hard to diagnose remotely, but I have two suggestions.
1) Make sure the plugin is enabled.
In Firefox goto Tools/Addons and look for Shockwave Flash plugin. Make sure it is enabled.
I think you can do the same thing in Chrome by typing about:plugins in the address bar.
2) If it is enabled, you can try using the official Adobe Flash uninstaller to completely remove flash then reinstall.
Official Abobe Flash uninstaller tool can be found here:
http://kb2.adobe.com...1/tn_14157.html
Reboot once uninstalled.
Download the latest version and try installing again:
http://get.adobe.com/flashplayer/
I hope this helps.