Some time ago, YouTube had switched to HTML5 videos for all modern browsers including Firefox. After doing some tweaks you can play all HTML5 videos in Firefox even without the Flash plugin installed. In some cases, e.g. when YouTube detects that something is wrong with your browser, it might switch automatically to using Flash for videos instead of HTML5, but there is no way to control this yourself. Let me show you a way to switch between HTML5 and Flash manually.
While Mozilla Firefox has good support for HTML5 standards, you might have noticed that it does not support all features required to play videos on YouTube without having Adobe Flash Player installed. It supports the required HTMLVideoElement, but some videos do not play anyway. Those videos require the Media Source Extensions feature which is disabled by default in Firefox. By default, this option is off in Firefox, so the browser is not able to play some HTML5 video streams. If you prefer HTML5 videos over the Adobe Flash Player, you might want to enable Media Source Extensions to get advanced support of HTML5 videos in Firefox.