The Xbox 360 HD-DVD Player
 
Initial Set-up
This thing requires the brainpower of a chimp to plug in and get it running.  You can read countless reviews with pictures of setup, etc., so I won't bore you.  The basics are that it has it's own (very small) power supply, uses USB to connect to your 360, and has 2 extra USB ports so that once it is is running, your 360 will still have a total of 4 USB ports open.  If you are using Xbox 360 Wireless Adapter, you can attach it to the HD-DVD drive instead of your 360 in the exact same way.
 
Quality of Picture and Sound
Sooo. . . is HD-DVD really better than watching a standard DVD?  The answer is: absolutely.  As you know, a standard DVD is 480p, the exact same resolution that standard-def TV uses.  DVD's are widescreen, but that is the only difference.  So, remember all those times you had to convince you wife/girlfriend/buddies that your new HDTV really is WAY better than your old tube?  All the same rules of engagement apply here.  HD-DVD is natively output at 1080p, if you have an older 1080i TV like me, it will appear to be exactly the same as 1080p to the naked eye.  Not sure what it looks like on 720p, but the Xbox 360 is famous for its built-in "ANA" scaler chip, so I'm guessing it looks really great at 720p.
 
Watching a movie on HD-DVD vs. DVD isn't quite as dramatic as flipping back and forth from HD to SD during an NFL game on Sunday afternoon, but the picture is much clearer and sharper (6X's more pixels than a DVD).  The brights are brighter, the darks are darker, all that kind of stuff.  *NOTE If you play standard DVD's in the player, they are upscaled and look MUCH better than when you play them in a standard DVD player.
 
The sound is phenomenal.  Basically, the Dolby Digital sounds a lot better because they are compressing it much less, and movies will have the new Dolby TrueHD format which the Xbox supports.  I never thought I would notice better sound, but it is definitely an upgrade over DVD. 

The Experience of Watching a Movie
 Watching an HD-DVD movie is way more fun than a DVD.  Why?  For starters, when you hit the menu button, the menus slide in over your movie, very much like your Xbox 360 Friends List, except that they are see-through.  So if you want to go to a different scene, check out an extra feature, etc., it doesn't require you to stop the movie.
 
The "extras" that are included on an HD-DVD are much cooler too.  There are standard "special features", but there new things like "U-Control", which can be many things, but usually seems to be a picture-in-picture that is sometimes synced with the movie and has someone explaining something about the scene, or sometimes showing the same scene from a different angle.  If enabled, it pops up while you are watching and gives you the option to show it or not. 
 

Does it beat BluRay?  Does HDMI matter?
Yes, it is better than BluRay.  Plain and simple.  I have watched a couple of BluRay movies, and they are basically no different.  In fact, it has been found by several high end audio/video sites ( www.highdefdigest.com) that the picture on HD-DVD is actually better.  The reason for this specifically is that they are using a better encoder to store the movies.  Apparently all the hype about storage space on HD-DVD being 30GB instead of 50GB is all for nothing according to these impartial sites.  Read this article if you want a comparison of Xbox 360 HD-DVD to PS3 BluRay: http://www.highdefdigest.com/feature_ps3vsxbox360addon.html
 
Does HDMI matter. . . tough one.  The answer is: only if you have a full 1080p HDTV, and even then not really.  Will you be able to watch a movie in full 1080p without HDMI?  YES, using VGA, which almost all 1080p HDTVs come with.  Actually 1080p works fine over component, it just that the TVs don't accept it.  There is nothing stopping you from taking advantage of HD-DVD today.  If you have a 720p or 1080i HDTV, then this is nothing you need to worry about anyway, component cables work fine for either of those sets. 
 

Is it worth the $$$?  Are there any HD-DVDs available?
The Xbox 360 HD-DVD Player is absolutely worth the $200. . . IF you meet the following criteria:

If you already meet those requirements, you really should pick this accessory up.  There are currently 150+ movies available and more are being released all the time.  In fact, Universal pictures alone is going to release over 100 new HD-DVDs this year, including some of Spielberg's classics.  You will not run out of movies to watch, every new release is released on one or both high def formats and DVD format simultaneously.  Since Christmas, I have watched probably 12-15 HD-DVDs and only 2 DVDs.  Almost every movie I want to watch is available on the new format.
 
Conclusion
I hope that this is somewhat informative and helps you understand that, for the money, this is a worthy upgrade for your Xbox 360.  In no way will it make your gaming experience better, it is just for movies.  If you were to buy a standalone HD-DVD player it would run you over $500, BluRay is close to $1000.  Neither HD-DVD nor BluRay is going away anytime soon, and the so-called combo players are insanely expensive ($1500 or so).  If you REALLY want to future proof, pick up a PS3 at some point and the format war will be over for you, but for right now, the HD-DVD is incredibly affordable, and there are TONS of movies available with hundreds more on the way

Amazon - Xbox 360 HD DVD Player

[Thanks KnottyMonkey]