Well, RPGs (especially for the PS2) are almost always single player. Two of the best games to come out for the PS2 in the last couple of years were Persona 3 FES and Persona 4. Then you have games like Final Fantasy X and Xenosaga I, II, and III. Each of which can be purchased for pretty cheap
FPS stands for First Person Shooter. Games like Halo and Doom
3PS stands for Third Person Shooters. Stuff like Gears of War and Syphon Filter
RTS is Real Time Strategy. Those are games like Command and Conquer, WarCraft and Star Craft
EDIT: Mulling it over in my head, if you want an RPG I think Final Fantasy X will be the place to start. Its accessible, has likable characters, graphics that still hold up pretty well due to fantastic art design, a great soundtrack and a great story. You can also get it for 5 bucks
I'll try and think of some good two player games for the system but to be honest I always felt as if the PS2 was weak in the multi-player area