phpBB, the software for this forum can embed videos. However, it expects youtube URLs to be a certain way and youtube will display a number of alternative URLs for the same video, which will not work in phpBB. However a few seconds of editing can fix that.
Basically, what you want is this:
You want the general URL for youtube followed by this string: watch?v=T6N3xkDjrkk
T6N3xkDjrkk is the code for that particular video.
If you get stuck, the easiest thing to do is to copy the title of the video at youtube, paste it into the search at youtube, go to that video and copy the version of the URL now in your browser location bar. Then just put the youtube tags to the left and right of it as pictured above.