If you want to embed YouTube videos in a post, click on the red TUBE icon in WYSWIG editor menu (may not appear if you are using quick reply or depending upon your UserCP settings). This will insert BBCode {YOUTUBE}{/YOUTUBE} (except that it will use brackets [, not braces). You can also just type the BBCode directly.
Then just insert the funky Unique ID for the respective video between the tags and away you go.
For example: For the above video, the URL from YouTube is:
http://youtube.com/watch?v=nzO9t7ibHgQ
The Unique ID is nzO9t7ibHgQ (the part AFTER the "=")
So, the code would look like:
[ YOUTUBE ]nzO9t7ibHgQ[ /YOUTUBE ]
Without the spaces in between the []. I had to put them there for the example so the software does not convert it to an embedded video.
Since the Video is hosted by YouTube and they encourage linking/embedding, we generally have no issues with copyrights. Copyright complaints are handled by YouTube, the person who posted the video, and the person who alleges ownership.