If you have a problem with Firefox and have Adblock Plus running as an add-on, you might have a filter that interferes with the Youtube player. I have Firefox 3.6.28 and Adblock Plus 1.3.10.
I discovered and deleted this custom filter:
The proper solution to solve the problem is to follow these stepts:
1. Enable the Adblock (APB) button in your toolbar, if it's not there already:
In Firefox: Tools -> Adblock Plus Preferences -> Options -> check "Show in toolbar"
2. Load Youtube and try playing a video. You'll get the black/blank screen. Click the APB button in your toolbar:
Click "Report issue on this page..." -> select "Adblock Plus is blocking too much" -> Then click "Disable" next to the filter(s) that cause the problem.
Now you can reload the page and hopefully see the video!