Well, SMT is a wonderful story that tries to depress you to death.
For reals though. It depends on the game... usually it involves something about something traumatic happening to the world and having to find a way to either survive and rebuild, or to discover WHY it happened to begin with. Have some cliff notes!
SMT! - You're a guy who gets a Demon Summoning program and gets involved with the end of the world. Afterwards, you side with a chaos cult, a kingdom of law, or try to forge your own way as you try to find your own way in the world. Very archaic.
SMT2! - The one that everyone remembers. Taking off after the neutral ending of SMT, you play as Hawk, a gladiator in some slums who eventually works towards creating a world of law, chaos, or human-ruled freedom. I don't DARE spoil the ending.
SMT If...! - Students at a high school are drawn into Makai because that one student everyone picked on decided to murder everyone with demons and goes kind demonic sociopath. It's actually the one that leads to Persona. I'm serious. Tamaki Uchida is the female main character in this game, and it's been shown that the 'Personaverse' spins off from her. The male main character's tale spins off into SMT: Khan, a rather odd manga.
SMT Nocturne! - The one that popularized the main series SMT series. The world ends, and you become a half-demon (Hitoshura) that works to rebuild it based on an ideal. Very, VERY grimdark.
SMT 4 - i'unno. Is not out yet.
And those are just the ones that were shown in the preview... there's Strange Journey, Devil Survivor, Raidou Kuzunoha/Devil Summoner, the main Persona series... there are a *Iot* of games under the SMT moniker. :)