When you go to the actual youtube website, do you get a message saying something along the lines that videos cannot be played due to java being disabled?
Also if you visit
http://www.whatismyflash.com/
does it say that your flash is disabled or not installed?
If this is the case then the version of flash on your computer is not installed properly and so you will need to go into the control panel on your computer and choose add/remove programs then click on and remove adobe flash player.
Restart the computer and then go to the adobe.com website and download the latest version again.