Megami Tensei ("Goddess Reincarnation"), know as Megaten for short, refers to both the 1987 NES game and the massive franchise that it spawned, though it should be noted that the whole franchise was really started by a novel series, Digital Devil Story (by Nishitani Aya), on which the game was based. Games falling under the Megaten umbrella to date include:
Digital Devil Monogatari: Megami Tensei - NES (1987)
Digital Devil Monogatari: Megami Tensei II - NES (1990)
Kyuuyaku Megami Tensei - SNES (1995)
Shin Megami Tensei - SNES (1992)
Shin Megami Tensei II - SNES (1994)
Shin Megami Tensei if... - SNES (1995)
Shin Megami Tensei NINE - X-Box (2002)
Shin Megami Tensei III: Nocturne - PS2(2003)
Megami Tensei Gaiden Last Bible - GB (1992)
Megami Tensei Gaiden Last Bible 2 - GB (1993)
Megami Tensei Gaiden Last Bible - Game Gear (1994)
Megami Tensei Gaiden Last Bible 3 - SNES (1995)
Megami Tensei Gaiden Last Bible Special - Game Gear (1995)
Another Bible - GB (1995)
Majin Tensei - SNES (1994)
Majin Tensei II: Spiral Nemesis - SNES (1995)
Ronde - Sega Saturn (1997)
Devil Summoner - Sega Saturn (1995)
Devil Summoner: Soul Hackers - Sega Saturn (1997)
Jack Frost no Meiro de Hiihoo! (aka Jack Bros.) - Virtual Boy (1995)
Megami Ibunroku Persona: Be Your True Mind - PSX (1996)
Persona 2: Innocent Sin (also known as Persona 2: Tsumi) - PSX (1999)
Persona 2: Eternal Punishment (also known as Persona 2: Batsu) - PSX (2000)
Giten Megami Tensei - PC-9800 (1997)
Shin Megami Tensei Devil Children - Red Book - GB (2000)
Shin Megami Tensei Devil Children - Black Book - GB (2000)
Shin Megami Tensei Trading Card Summoner - GB (2001)
Shin Megami Tensei Devil Children - White Book - GB (2001)
Shin Megami Tensei Devil Children - Light - GB (2002)
Shin Megami Tensei Devil Children - Darkness - GB (2002)
Shin Megami Tensei J - Mobile phone sevice (2002)
There are numerous ports as well that are not included in this list.
The series has also resulted in two animes: one movie also called Digital Devil Story and an anime series based on Devil Children.
The two defining points of the Megaten series are the ability to interact with 'demons', and the occult themes that tend to run through the games. It should be noted that in the Megaten universe, the word 'demon' applies to pretty much any mythological/religious entity. Everything from the seraph Michael to Vishnu to Baphomet is classed as a demon. And, if it's a demon, chances are you can try talking to it or summoning it into your party.
As for occult themes, this tendency is inevitable given the game's focus on demons. Plot summaries for the games listed above would blow this node to astronomical proportions, so please see Shin Megami Tensei to get a good idea of how Megaten plots tend to go.
Atlus is the company responsible for pretty much every Megaten game although Namco produced the first two Megami Tensei games, and Enterbrain were responsible for Trading Card Summoner which was based on the Shin Megami Tensei trading card game.
The series is quite popular in Japan, but is more of a cult interest in the West due to the rather low amount of translated games.
Sources:
Velvet Room (http://www.geocities.co.jp/Playtown-Denei/1421/index.html)
GameFAQs (http://www.gamefaqs.com/)
Thanks to Re-Miel for some corrections and Shadowsithe for reminding me about the anime.