Can you go to a video and try this? In the URL add at the end the following (without the quotes): "&html5=1"

It will force youtube to serve HTML5 video, in order to confirm if you got a flash plugin issue.

No dice. Just a black screen, and then it stopped responding. And then I found out it doesn't work with Vimeo either.

That's really weird if it happens on two different sites serving video content. Sounds like a driver or an addon issue. Can you check "about:addons" and disable the ones related to flash and/or unecessary ones? Ultimately you should consider updating your video drivers, just in case.

And what about a pure HTML5 video liek this one? http://www.craftymind.com/factory/html5video/CanvasVideo.html

Well, the HTML5 video works. But I don't seem to have any flash plugins or anything. And I've just tried DailyMotion and THAT'S run into problems, too! So, that's Youtube, Vimeo, and DailyMotion that's not working. Something is royally borked here. BTW, I'm using Firefox 43

If you didn't see the flash plug-ins, you probably just need to install Flashplayer: https://get.adobe.com/flashplayer/

cfcentaurea said

If you didn't see the flash plug-ins, you probably just need to install Flashplayer: https://get.adobe.com/flashplayer/

... I can't believe I overlooked that detail.

Happens to all of us. :-)