Want to watch YouTube videos in the best available quality? There is an option for that. As Alex Chitu showed, you can just click on the drop-down menu for switching between formats, click on "Settings" and enable "Always play HD when switching to full screen (when available)". You can also change playback quality settings from this page.
Unfortunately, when you exit the full screen mode, YouTube doesn't switch to the previous format. As YouTube's help center explains, "if you play HD video in a small player, the computer works overtime to scale down the video to fit within the player, which may result in choppy playback. It's always best to play the video size that best fits the size of the video player."
So the best option will be to manually set the video size for each video you playback. Playing HD quality video might not be the best solution at all times.