Try another browser and see if it works. Chrome, Firefox, Opera, Safari.
If it does, then you know you just need to get a patch/plugin/extension for your current browser... unless you are saying your whole screen goes black. It sounds like you are saying you can see the rest of youtube but not the stream.
I can view videos on youtube, mostly anyway. when people post the same youtube videos on my facebook, when I play them directly on facebook all I get is a white screen but video is playing,but I cannot see pciture, when I open the actual youtube I can see the video, the same thing happens if I watch videos on some sights, say like MSN, video is playing but all I have is a blank screen. tried re-loading new adobe, seemed to have been suggested somewhere - to no avail.