If you're looking for the "next gen" video experience (read: not using flash) and use Youtube, you can hit the YouTube HTML5 page to add your account to the trial. Note that not many browsers are modern and awesome enough to support HTML5 properly, so you are currently limited to what you can use:
Right now we support browsers that support both the
Sadly a few restrictions are in place, mostly that if the video has ads in it (more and more these days :( ) it'll play in the flash player, and no fullscreen. However, I expect the option to get better and better as time goes on. It's also not perfect. For example the volume control seems a bit touch in terms of allowing you to adjust it while the video is playing.
If you're a Google Chrome user you can also check out the Youtube HTML5-ifier extension to do something similar.