Persona 1 and 2 were pretty different, the inflection point in the franchise is 3 (also when it exploded in popularity). So when I say "Persona" in this post I mean "Persona 3+".
Persona basically combines JRPGs, a bit of rougelike, a bit of Pokemon, a bit of "raising sims" (like Princess Maker), and a bit of VNs.
Let's unpack each of these:
- JRPGs -- this part should be obvious. The turn-based battle system, character leveling, equipment, and dungeon exploration are pretty vanilla JRPG.
- Roguelike -- the dungeons have randomly generated floors and change each time you visit. You also have rare encounters and drops and things like that.
- Pokemon -- your character can use the titular "Personas". You can aquire them in-game, improve them, and fuse them in a pretty neat system to create new ones. Much of the strategic interest is in preparing Personas well suited for particular battles. (Note, in terms of influence, this should really be "SMT", but a lot more people know Pokemon)
- Raising Sims -- the entire game is based around a calendar. On each day (and sometimes time slot), you can decide what to do. This can be anything from studying or working various side jobs over hanging out with other characters or eating out to actually going to a dungeon. All of these activities can influence, and are in turn influenced by, your non-battle skills (like knowledge, courage or diligence).
- VNs -- over the course of the activities I just described there are a lot of character stories you get to participate in, and you have quite a bit of choice in them (though ultimately the most meaningful choices you make are how you spend your time).
So, take all that, combine it with a modern-day Japanese highschool + supernatural setting and
extremely confident, consistent and unique overall graphical styling and music, and you get Persona.
Sadly, it doesn't always work out perfectly, and there are pacing issues especially with 5 and dungeon and battle "design" issues particularly with 3, but it's also clear that there are reasons for its rather unique popularity/appeal.