RPGs
Sonic and SEGA
Anything with a good plot/story.
You NEED to check out Yakuza 1 + 2. They're a series of semi-open world beat 'em up RPG hybrids made by SEGA. Think Shenmue but with more to do, better controls and (IMO) a better story and characters, even if the cut scenes have a tendency to get rather long winded.
Other than that, I would highly recommend Persona 3 and Persona 4. IMO Persona 4 is the superior game, with much better game play and narrative, although 3 is still worth your time.
Final Fantasy X and XII are also really good, X is more linear with rather strategic turn based combat while XII is more open with (IMO) somewhat clunky but still good semi-real time combat. Both are good, but its apple to oranges.
Xenosaga isn't terrible either, I've only played the first one and its got a neat combat system and an interesting story (think 2001 but with less subtly and with anime tropes out the ass). Although it is really, really flawed and the cut scenes drag on forever. Its cheap these days so if the game looks interesting to you I'd suggest giving it a shot.
Star Ocean: Til the End of Time isn't terrible. The story sucks IMO but it has a fun combat system. Just know that if you're going into the game expecting a sci-fi RPG prepared to be very disappointed.
Those are just a few RPGs off the top of my head for the console that you might dig. If I remember any more later I'll come back and edit them in.
EDIT: A few non-RPG suggestions that you might like:
The Jak trilogy is a series of platformers with the first game being rather traditional while the second and third games mix up the platforming with vehicles and gun play, the narrative is actually pretty interesting as far as platformers go to boot. IMO best game in the trilogy is probably 3, but 2 left more of an impact on me.
Shadow of the Colossus is practically a must if you own a PS2, the whole game is a series of boss fights where each fight is a puzzle. Easily one of the best games on the console IMO.
Metal Gear Solid 2 + 3 if stealth game play with a complex plot (although the cut scenes are really long and really drag in places) appeals to you then I highly recommend these games. IMO MGS3 is the better of the two games, with a significantly better game play that also focus on survival and a better story.
There's a few other games I had in mind, but most of them likely wouldn't appeal to you. Plus this post was starting to go from PS2 recommendations to Finalark's Favorite PS2 Games.
Persona 4 is a must. Priority number one for you.
Persona 3 is great too, but the gameplay is a bit of a step back from P4.
Goddamn it, Rutger, you beat me to it. Well, Mari, now you have two people's word that P4 is good, so consider that.