Hi William and others,
If you can install add-ons, are stuck on Moodle 2.6, and really need to show YouTube videos, you can do it with the Generico filter instead of the YourTube Repository. For ease of use, anyone on Moodle 2.7 or higher will also want to add the Atto: Generico button. (Sadly, the button files will install in Moodle 2.6 but the actual toolbutton will not show up on the toolbar.)
There is no convenient way to exchange Generico filter settings, so here is a picture:
Here is what you would need to type into the editor of any place you want to show a video, such as a forum, label, topic, etc.
And of course Moodle 2.7 and above can make it easier by doing the typing for you because it will let you setup and use the Generico button for the Atto toolbar.
This works so well, I have not used the YouTube repository since Justin Hunt released the Generico filter. What's even better, you can set up filters to do many other nice things too. For instance, my site, quizzes, activities, and etc, all greet my students by name.