If you want to play a YouTube video one more time, there's a "replay" button at the end of the video or you can press the left arrow, after focusing the video.
To automate the process and play the video in an "infinite" loop, use Loopy for YouTube, a Greasemonkey script that adds a small link below the video to start the loop. The script requires Greasemonkey , a Firefox extension, and it's efficient: it doesn't reload the page to download the video again.
There's another way to play a YouTube video in a loop - use the loop parameter of the embeddable player (here's the complete list of parameters):http://www.youtube.com/v/VIDEOID&loop=1 (replace VIDEOID with the video's ID)
If you add the autoplay parameter, the video will start to play automatically:
http://www.youtube.com/v/VIDEOID&loop=1&autoplay=1