What exactly is Shin Megami Tensei, or Persona? It's simply an RPG game where you control various Personas to achieve victories in various dungeons. I don't know what the premise or overview is in the first three games, but I know for a fact that you can control various amounts of Personas from fusing them or beating up enemies.
Persona 4 is easily the best Persona in the series, and not only that, probably one of the best RPGs on the PS2 if you're able to play mature games. It covers various (slightly disturbing?) themes and has a random dungeon system. The dungeons themselves range from a castle to even a strip-club. Just think of the bosses...

There's also a Social Link System which works similarly to a Dating Sim in some cases. The more you befriend characters, the more Personas you can make with power, and a bonus can be granted too. :3

Can't get any better screenshot though...Unless this counts:

You can discuss other Persona games here...And you can discuss the dirty Personas if you must. O.o