I've never successfully watched a video on Twitch. I get a black rectangle with playback controls, and when I press play, nothing happens. Disabling AdBlock doesn't help.
I don't have Flash installed at all. Does the site require Flash?
If so, then that's news to me; other video sites that require Flash usually a) show a "you need Flash" message in the place where the video whould show, and b) don't show playback controls, because they're part of the Flash component itself. Also, I never saw any mention of Flash in any of the site's help/troubleshooting/FAQ documents.
Since posting my last message, I looked at the documentation on the website again, and saw that it claimed that you could use the site on iOS and Android by just using an ordinary browser. So I tried visiting it in Safari on my iPhone, and the videos worked. Then I tried using Firefox on my Kindle Fire, and the videos didn't work. But they do have a dedicated Twitch app for the Kindle, so I downloaded that. So now I have a way of watching Twitch videos. :-)
The fact that Flash is not mentioned anywhere on the site as being a requirement seems like a glaring omission.
"ordinary browser" = Chrome for Android, latest version, Safari for Mac, Safari for iOS
Firefox does not support HLS in either desktop or mobile version. It requires Flash as a last-ditch fallback on any platform that doesn't support HLS, afaik.