I downloaded a few videos online. They were free. But my question is, they have a watermark dead center of the video. I wanted to know if there's a way that I can alter them to remove this as it kind of gets in the way of the videos.
Nope; there isn't. You will either render the video unwatchable or otherwise crap looking. You could blur over it I supopse, but as I said, it won't look great.