I've been struggling this with issue for about three weeks now, uninstalling and reinstalling the most recent flash update. I've tried a few of the older versions as well to see if that made a difference and the videos started working for a day or two and then immediately went back to the same black screen on Youtube. Videos hosted on other sites that are not embedded Youtube videos work just fine. At one point I completely uninstalled flash from my computer and surprisingly, youtube videos started playing without any problems even though Firefox continually asked me to instal the flash plugin. I wanted to play videos on other pages and thought that maybe this time around it wouldn't interfere but now the videos won't play, again. I have the newest version. I don't know if the bug is with Youtube, Flash or Firefox but it's getting pretty ridiculous. This didn't happen at all about a month ago so I'm wondering why all of a sudden I'm having this issue and how I'm supposed to go about resolving it.
Also, at one point I tried another browser and downloaded Opera and the issue persists there as well.