If I were to take videos from Youtube or Break, and place my own domain name on them before placing them on my website, would that be illegal? What about immoral? or is this just no big deal?
Well, in my view it would be both illegal and immoral. It's definitely illegal since you don't own the copyright of the video and you don't have the permission from the copyright holder to use the video on your site. Adding your own domain name to the video implies that you created the video or own distribution rights which isn't the case.
I agree with Crusader. You are opening yourself to great legal implications by doing this. Not from Youtube, but from the owners of the video.