Well I've had my XBox360 for awhile now and I have all of the highly rated games (on gamespot) as well as a few lower rated arcade games. I'll give my own personal review of each one here.
Call of Duty 2
The gameplay puts you, a lone grunt on the battlefield, in the middle of World War II where you play as a Soviet, a British and an American soldier. For all those that enjoy first person shooters I'd highly recommend this one. The graphics on the XBox360 version are incredible and not in any way substandard to the PC version.
Perfect Dark Zero
Another first person shooter that I'd highly recommend. The gameplay puts you, Joanna Dark (And if you so choose, your companion on the other controller), against an organization bent on global domination. The graphics are superb and the storyline is excellent. If you like the recent James Bond games you'll love this one.
Need for Speed Most Wanted
I've been a Need for Speed fan for quite awhile now and there is no way that this one was a disappointment. Gameplay puts you against various blacklisted racers around the city in an attempt to get your original car back. Not only does the blacklist stand in the way of your prized car but the many cops driving throughout the area are very intent on bringing you down. If you're looking for realism you'd probably have more fun with something like Project Gotham Racing 3 however if you're looking for pure fun and excitement this is the game for you.
Project Gotham Racing 3
As stated above, realism is the key to this game. You drive one of eighty different cars in Formula One styled races in various cities around the world in an attempt to reach the top. The graphics, in my opinion, are the best in the group of titles that were available for the 360 at launch. The in-car camera view has to be seen (and played) to be believed. Another highly recommended game.
Kameo Elements of Power
If you're looking for more of a family title that both you and your kids could enjoy Kameo won't be a disappointment. Gameplay pits you, a princess able to transform into various elemental creatures with different powers and techniques, (And if you choose, your partner on the other controller) against the troll king and his minions in an attempt to rescue elemental ancestors.
Bejeweled 2 Deluxe
Downloadable from the XBox Live Arcade, Bejeweled 2 is an addictive gem-swapping game in the same fashion as the original. This time around you are given the task of unlocking new playable modes while attempting to achieve the best score possible so as to get highly ranked on the Live Arcade leader boards. If your looking for a time-wasting game that won't disappoint this one comes highly recommended.
Another game downloadable from the XBox Live Arcade. Zuma is an addictive puzzle game where you have to survive the tribulations of various temples in order to achieve the best score possible. You're given the option of playing in one of two different game modes, each with their own highscore list on the Live Arcade leader boards. Made by the same company that makes Bejeweled 2 and, while not as addicitive as Bejeweled, can still hold it's own against most of the other arcade puzzle games online.
Hexic makes it's jump to High Definition with the release of Hexic HD, a game provided for free with the purchase of an XBox 360 Premium System or an addon hardrive for the Core System. Gameplay consists of matching same-coloured tiles in order to remove them from the board and perhaps scoring power tiles. Not quite as graphically intense as Bejeweled or Zuma but still highly addictive.