Shin Megami Tensei is the main series and the franchise on the whole, and can probably be described as "a far more complicated Pokemon, only with demons, death, a lot of darkness, and a stupendously high difficulty level." Rather than catching adorable little creatures and using them in largely consensual battles, you're using demons/angels/mythological creatures to defend yourself from other demons/angels/mythological creatures, which are trying to rip you apart. And usually joining in yourself with swords, guns, and magic. And then usually dying, because it's really hard.
It's a pretty bleak series. God is definitely not good. Lucifer is... usually not that bad, instead embodying freedom (and all the horrors of anarchy that can ensue). If you look deep enough, there are a lot of philosophical references and ideas in there. Or you can just summon demons to hit other demons. Still, if you're deeply religious, be warned that it might well offend.
Persona is one of the many, many spin-off series, and while usually dark, it's also a lot less weighty than the main series. I'm not going to go into the first two Persona games, but Persona 3 and 4 cast you as a high school student who has to deal with day-to-day life (going to school, passing exams, joining clubs, making friends) while fighting monsters by night.
The Persona games rely a lot on side-characters and a mini social-simulator, and that really makes them stand out. The fact that they're tremendously well-written and have a fast and well-made RPG battle engine also help. They're still brutally hard, but a lot less complicated and a lot more "fun" than the main series games. They're often laugh-out-loud funny and have a lot of fun with the concept of high-school students being the only thing standing between the world and the apocalypse, but they also take their subject matter pretty seriously.
Persona 4 has also had a pair of fighting games, which are made by Arc System Works (Guilty Gear, BlazBlue) and thus are really, really good.
I've been over this in another thread recently, but if you want to get started with the series, Persona 3 is probably the best place to begin, in any of its many releases.
TL;DR: Shin Megami Tensei is Pokemon that'll make you question religion. Persona is a spin-off series about the joys of having to balance fighting monsters with going to school. Both are pretty hardcore JRPG series, but Persona is by far the simpler and more accessible of the lot.