Persona is a part of the Megami Tensei series. The main series is Shin Megami Tensei. Staples of the main series are the demon summoning, demon negotiation, demon fusion, the Law and Chaos conflict, demon apocalypses, high difficulty, and many other things.
Tamaki Uchida is from Shin Megami Tensei if..., which is a side story to the main Shin Megami Tensei games (it's also a main game). It takes place in a timeline in which the events of SMT 1 are averted, where the demon invasion was fought back by Devil Summoners ( which exist thanks to Devil Summoner on PS2), Japan was never nuked, and the instigators of the events in 1 are arrested/killed (the prologue of SMT 1 still happens though).
In if..., Tamaki's school is sent into the Expanse, or the demon realm, and she must partner up with a classmate to get out (who she chooses determines the ending).
Stephen, the creator of the Demon Summoning Program from SMT 1 and 2 (and 4/Apocalypse), sent the program to Tamaki's school. That's how she gains the ability to summon demons.
Persona takes place in the Devil Summoner/Persona/if.../Catherine timeline branch. In 208X, during the events of SMT 2, a man named Raidou Kuzunoha the 40th decides that life in Tokyo Millennium sucks so he travels back in time to the Taisho era of Japan to prevent the nukes from ever launching in 199X (early nineties).
Raidou Kuzunoha the 14th (Not a real name, just a title) is a Japanese high school student and next in line to become the new Raidou Kuzunoha, the choices he makes in Devil Summoner: Raidou Kuzunoha vs King Abaddon (PS2) determine the condition of the world (Law/Neutral/Chaos). Siding with Law makes what Raidou Kuzunoha the 40th did futile as it causes the timeline to repeat and the events of SMT 1 aren't avoided. Siding with Chaos leads into SMT 3: Nocturne (and maybe 4 and Strange Journey). And finally, Raidou the 14th being Neutral leads into the Devil Summoner/Persona/if.../Catherine timeline.
