When I go to the mobile version of YouTube, I see icons of videos that I can try to view but none of them work - the video tries to load and then always errors out with Cannot Play Video - Sorry, this video cannot be played (sure would be nice to have the option to get a more detailed explanation of the error). . .
I just got the NookColor so I have not had the chance to test copying videos onto it and seeing if they play or not. . .
This has been asked many times already. Search the forums and you'll find the answer.
Hint: NOOKColor doesn't support flash (yet) ...
That's right nook color doesn't support flashplayer yet and youtube videos need flashplayer to play.
The mobile version of Youtube doesn't depend on Flash. The "old" m.youtube site uses an RTSP link to a 3GP version of the clip (which your default movie player plays), while the "new" m.youtube site uses HTML5. You'll know if you're on the "new" site because it has the more contemporary UI with like/dislike, instead of the star ratings (you can also look at the page source for the <video> tag, or lack thereof).
FYI, the Nook Color still accesses the "old" site. Apparently the browser doesn't have the required codecs (h.264 or WebM) to be able to access the "new" site.
And as far as why the "old" site is not working for you, here's some info from someone in authority: