I had this problem, which occured suddenly without warning. I did try some of the suggested remedies. But the problem was that although I seemed to be able to successfully download and install the relevant Adobe Flash 10 plug in, my Firefox did not acknowledge
it and it was not shown in the add on section of Firefox. However I downloaded the Flock browser from the get Deb repositries and that now works in that I can play you tube videos. I did try other browsers, such as the Mint version of the Epiphany browser but I could not get that browser to work at all ,and I had similar problems with the Gecko-mozilla browser.