Up until the last year or so, browsers have always required Adobe Flash Player to display Youtube videos. Since Google updated Youtube though, many videos will now display using HTML5. Support for HTML5 commenced with Internet Explorer 8 but didn't really become part of the browser's capabilities until IE9. However Flash Player is still needed to display some Youtube videos with all versions of Internet Explorer.
So, as Tony suggests, you might need to install (or update) Adobe Flash Player. Although Softonic (in Tony's link) is normally a reliable source of software, you should only ever download Flash Player from Adobe's own site, here:
https://get.adobe.com/flashplayer/
(NB: Remove the tick under 'Optional Offer' before clicking on 'Install Now')