The first option requires basic knowledge about Youtube’s url structure especially the unique video code that is part of every url. It is possible to change the url of the Youtube video so that it will loop and autoplay automatically. Here is how you can do that manually:

A normal Youtube url looks like the following

http://www.youtube.com/watch?v=sgjMK2sWgsE

You can turn this into a replaying and autoplaying video url by appending the parameters loop=1 and autoplay=1 to the url and changing the url slightly so that it looks like this afterwards

http://www.youtube.com/v/sgjMK2sWgsE&loop=1&autoplay=1

This will open the youtube video in fullscreen in the web browser, play it automatically and loop it infinitely.

The manual solution to loop Youtube videos is fine if it is only used scarcely. Users who want to loop Youtube videos regularly might want to consider an automated option like the Youtube HD Ultimate userscript which comes with the functionality to automatically loop Youtube videos. Other userscripts are available that offer a similar functionality.