If you like Final Fantasy type games, then I do not suggest Final Fantasy XIII, basically everything Galaxia says above is why. HOWEVER, I do suggest Lost Odyssey. My BF is a huge RPG buff and really enjoys the game because it feels like some of the older Final Fantasy's and one of the better RPGs on the system.
It would help to know what kind of genres you like?
Well ignore that and just read all of this huge post as a general over view of a good games from a variety of genres.
For fighters, Mortal Kombat is hella fun, they made the game really great is a great sorta reboot of the series. I haven't played Marvel vs Capcom 3, but I've heard very good things.
For action/button smashing, Bayonetta is very good. In fact its one of my favorite titles for the Xbox360. If you've played any Devil May Cry game, or at least know what the game play is like and enjoy that type of game, Bayonetta is fun. Its campy and the characters are really over the top, but it adds to teh charm of the game, plus the game play is really fun, never boring. The story is confusing as all heck, but honestly it warrants a replay through just for playing on the harder difficulty settings and getting the achievements for fun. There is also Devil May Cry 4, i don't remember the reviews on the game but since its older its shouldn't be very expensive anymore.
For open world sandbox games, Grand Theft Auto IV and Red Dead Redemption are two games that I think of. I didn't play GTA, but I did play Red Dead Redemption. Overall, its a fun game, especially since its basically GTA but in the wild west. You can basically go around anywhere on a horse, steal peoples horses, kill people, tie up whores and stick them on railroad tracks (there's an achievement for that btw), however the missions can feel repetitive at times, but the story is very good.
Open world RPG: there's Fallout 3, Fallout New Vegas (which is standalone from FO3 so no worries about needing to play one first, but i recommend Fallout 3), Mass Effect 1 and 2 (YOU GOTTA PLAY THOSE THEY ARE GREAT), Elder Scrolls IV: Oblivion. You can easily waste hours and hours and hours into any of these games. Elder Scrolls will give you the fantasy, Mass Effect the sci-fi, and Fallout3 the post-apocalyptic sci-fi.
To give a little more detail about the Mass Effect games, basically its an RPG with 3rd person shooter elements. You play as Commander Shepherd who is out to save the galaxy (i'm not gunna give too much away). You can choose who he (or she!) looks and you can level up the character any way you want. Also, depending if you decide to be Good, Renegade, or Neutral will depend on a lot of the outcomes of the game. Basically, you command this ship, The Normandy, in which you have a crew, many of them you find along the way. You can chose who you want to go with you on missions, with up to 2 of your crew. You can level them up as you choose. Have relationships as you choose, etc. I haven't gotten to Mass Effect 2 yet cause I'm still on the first one, but from what everyone has told me, its only improves on Mass Effect 1. So wooo boy, i need to get there. lol.
For Zombie games: Left 4 Dead 1&2, Dead Risings 1&2&2.5 (it just came out), Dead Island. L4D is your typical 1st person shooter with zombies, but it focuses all on co-op with up to 4 player online co-op. Its hella fun and insane and its made by Valve so thus its good. lol. I haven't really tried Dead Rising, but its more open world ish and instead of being strictly a shooter, you can pick up items from around you and make weapons out of them. You can do some pretty fun stuff with that.
Dead Island is like a mixture of Fallout and Borderlands (which i talk about in a bit). Its an open world Zombie game with missions and items to pick up and collect, etc. etc. Its a pretty huge map (you're on a resort island) and the co-op is pretty fun. I haven't had a chance to get it, but when it came out, we were sold out of it at work for a few weeks. it was the popular.
For shooters: any of the Call of Duties, Battlefields, Halos, and Borderlands are all the ones I can really think of off hand right now. I enjoy Halo as being the basis of what we now know of first person shooters. I always have a blast with co-op and multiplayer modes, but i don't play it that often (rather at all since i don't own it, but if i did, i'd play it lol). Call of Duty is straight run and gun type game. I haven't played Battlefield, but from what I understand its more indepth of a game (more strategy than run and gun type games) and has destructible environments. Cool. Now Borderlands. Borderlands is pretty awesome. I've only put a couple hours into it, but it was a hella fun two hours. Its basically a shooter with RPG elements and a item drop system similar to Diablo II. You can play co-op (imo its more fun that way) and you can choose from 4 different characters. They each are their own class and have different powers to use besides shooting. You can also level them up and upgrade the stats, weapons, etc.
There are also some fun puzzle/casual games too like Plants vs Zombies (which comes with Peggle and Zuma), and uh...crap i forgot the others i was gunna mention. /cough
Also, I highly recommend any of the Lego games, espcially the Star Wars ones. They are a good all ages game and just fun all around. Its jump in and out 2 player and got a good share of puzzles and beat up lego action. I also like the Harry Potter one too! In all they have Lego Star Wars Complete Series (covers the movies 1-6), Lego Star Wars III: Clone Wars (covers Clone wars cartoon), Lego Batman, Lego Indiana Jones 1&2, Lego Harry Potter Years 1-4, and Lego Pirates of the Caribbean. Yes it seems they are legoing everything. But why not? It's legos!!
I would suggest some Xbox Live Arcade games, but since you don't actually own the system, I figure you probably won't be getting any. However, if you want some suggestions, feel free to ask.
Oh another RPGish game I forgot to mention was the Kingdom Under Fire game that came out for it. Its pretty fun. Also, Hunted: Demon's Forge isn't too bad and Dungeon Siege III is a pretty all right dungeon crawler type RPG. Similarish to Diablo II, but i think more linear.
Anywaaaay, I hope that helps. I tried to think of as many as I could off the top of my head. However, if you tell me more of your tastes in depth, and if I see anything while at work that sticks out in mind, I'll try to remember to tell you abou tit. Also, from what I understand, there are games that get released over in Europe that don't always get released over here and stuff. Like the UK got Doctor Who games and stuff (I WANT DOCTOR WHO GAMES WTF). So um, i won't be knowledgeable of any of those.
Ok, sorry this got kinda TL;DR but I got into "Brittany's at work" mode and yeah. And it basically went like this: