SharePoint 2010 solution
- Open youtube video in a browser, and copy video ID from the URL bar (e.g. for http://www.youtube.com/watch?v=QhyYHlB38wg ID is QhyYHlB38wg).
- Insert video ID between http://www.youtube.com/v/ and ?version=3&feature=player_detailpage and compile the result link (e.g. of a result link http://www.youtube.com/v/QhyYHlB38wg?version=3&feature=player_detailpage)
- Put cursor into the question text area, Use Insert tab of the Ribbon and choose Embedded Media > from Address> Flash> specify the window size and paste the link into the video address.
SharePoint 2013 solution
- Open youtube video in a browser, under the video frame click Share button, select Embed and copy the embed code (e.g. <iframe width="560" height="315" src="//www.youtube.com/embed/QhyYHlB38wg" frameborder="0" allowfullscreen></iframe>) in the clipboard.
- Put cursor into the question text area, Use Insert tab of the Ribbon and choose Embed and paste the embed code in the field appeared and click Insert button.
NOTE: Some video authors restrict playing videos on other than YouTube sites.