When you watch a YouTube video a temp file containing the video is actually downloaded onto your computer. It is in the Temporary Internet cache. If you can find it, rename it and move it to a more permanent location and you will have the video. When looking for it, sort the files by size and look at the big files.
If you just need the video for viewing later, you can let the video load completely and then not close the browser. Later (not more than a day or two later) you can play it again even if you do not have internet access.