This has nothing to do with your computer hardware.
The problem is inside the ADOBE FLASH. At last update they brake it... is same everywhere. if you have an old flash offline installer use it and it will work again.
It is not quite correct - I use Firefox with Adobe Flash Player, with version 11,4,402,287 installed and I can watch any video without a problem.
The point is that you have to download Adobe Flash Player version compatible with your internet browser http://www.adobe.com/software/flash/about/
Assuming you have the correct version, and your internet connection is good enough for download YouTube videos, then the problem might be your hardware -- by the way what is you graphic card?