This depends on the play mode that is set when you are logged into youtube on Firefox.

What plugin loads, you can also control auto play in the browser if this has changed.

Thanks Guigs, but this doesn't help. I never log into YouTube. I only view the videos through other websites such as my homepage or Facebook. Where would I find "auto play" in the tool bar and what would I want it set at? Thank you very much.

A quick check to make is running Firefox in Safe Mode to see if an Extension, Theme, hardware acceleration or some other assorted settings are at fault. https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-using-safe-mode

Since you list Youtube, the videos in question should be on HTML5, but a plug-in check never goes amiss: https://www.mozilla.org/en-GB/plugincheck/. (note: plug-ins are not affected by Safe Mode)

If the issue persists even in Safe Mode and your plug-ins are up to date, then the next things that come to mind that would target Firefox are antivirus and malware.

Updating graphics drivers is usually not a bad idea, but might be an unlikely fix since IE can play the content fine.