By default YouTube will want to play videos in flash format, requiring Adobe Flash Player for playback. For several reasons, not in the very least for security reasons, usage of the flash player should be avoided.
1) Tell YouTube to present videos in HTML5-format.
- Go to youtube.com/html5
- Here, hit "Request the HTML5 player"
- Do not close the tab
2) By default Firefox may not play at qualities higher than 360p. Fix.
- Open another tab
- Type about:config
- Using the search-bar, search for each of the following entries and make sure they are set to True (double-click to toggle)
(If you cannot find an item, just ignore it)
- Go back to the first tab (youtube.com/html5) and take a good look at the items at "What does this browser support?"
- Press the circular arrow in the address bar or press F5 to reload the page
- Many more items should now change from a red exclamation mark into a blue check mark
That's it! Now you can enjoy YouTube videos up to 1080p.