In order to play videos in a browser you need some form of video engine. Adobe Flash and Microsoft Silverlight are the two I'm aware of, Google may have it's own (I would have no idea). What engines are supported is site specific. Generally if you don't have what the site wants it will offer a link to install it.
I can see youtube perfectly fine in IE8, 32-bit version w/ flash installed, and in Firefox, 32-bit w/ flash.
There is a beta version of adobe flash available for the 64-bit IE browser but to my knowledge the production version has yet to be released.
32-bit browsers work perfectly fine in a 64-bit OS and I would guess-timate that currently 99% of W7 users use a 32-bit browser.