YouTube videos require a software package called "Adobe Flash Player". If you go to Google and search for it, you should be able to download it and install it for free.
Another thing to check is this: Does a thin message at the top of your Internet Explorer window show up with the message "Pop-up blocked. To see this pop-up or additional options, click here...."? If it does, click this message and select either "Temporaily allow pop-ups" or "Always allow pop-ups from this site...". Only choose the second option, if the site is reputable.
I hope this helps.