YouTube has been working on its HTML5 video player for more than a year now, but the project has remained mainly experimental. Still, the latest versions now seem rather complete and on par, feature wise, with the regular Flash-based player.
As Google Operating System noted, the HTML5 player is pretty much ready to take over the Flash based one. That said, that probably won't happen any time soon.
Google could decide to start advertising the HTML5 version to users and start offering them the option to switch, but that depends on how stable the player and, importantly, how well browsers out there handle it.
The HTML5 YouTube player is available as an experiment. If you decide to join, all YouTube videos will be streamed via HTML5, except the ones with ads. Of course, whether this works or not depends on your browser.
But if you're using a fairly modern browser there should be no problem. You can always leave the HTML5 experiment, of course.
The HTML5 player looks rather well, in fact, in some respects, it looks better than the Flash based one. It also offers the same functionality, all of the controls are there, you can enable subtitles (close captioning) and there's even full screen support in some browsers.
What's more, you'll also be able to get videos streamed in WebM, Google's open source video format, rather than the usual h.264. WebM videos are available in 480p and even 1080p versions on YouTube, depending on the original resolution of the video.
Google made a lot of noise last year with the release of the WebM video format and it's always been a big proponent of HTML5. But things haven't really changed in the meantime. The HTML5 player is a lot better, but it doesn't seem like YouTube plans to push it to users any time soon.