You want an introduction to Final Fantasy? In 1987, a game company called Square was on the verge of bankrupcty. The director, a man called Hironobu Sakaguchi, termed the company's last venture his "final fantasy". That game was such a hit, they were forced to make a sequel. And thus began one of the longest running and most ironically titled franchises in videogame history.
Final Fantasy VII was released in 1997 to unanimous applause. It is the seventh in the now widely-popular Final Fantasy series. Because of its enormous popularity, Square-Enix has decided to expand on the game in a series of follow-ups.
The actual game follows a cold mercenary and his rag-tag group of liberators as they try to save the world while at the same time learning important lessons about themselves. First they fight against an evil power conglomerate known as Shin-Ra, befor the greater threat reveals itself in the manipulative alien JENOVA and her "son", the ambiguously gay badass Sephiroth. The hero and his party, of course, defeat the both of them, but not after suffering some casualties of their own and the most confusing game ending known to man.
Taking place some years before the events of the game is Before Crisis. It's a cellphone game currently only released in Japan, though they've promised a stateside release sometime in the future. It features the Turks, a mercenary band for the aforementioned evil power conglomerate. They're out to stop AVALANCHE, an eco-terrorist group with the emphasis on terrorist, not to be confused with the similarly named group of do-gooders you control in the original game.
Final Fantasy: Last Order OVA is a short anime film chronicling the Nibelheim incident, where Sephiroth went nuts a few years before the start of the game. It's a scene recounted well in the course of Final Fantasy VII, except now it's much, much prettier. It was released as an extra in the special "Limited Edition" version of Final Fantasy VII: Advent Children.
Final Fantasy VII: Advent Children is the much-anticipated movie sequel to FFVII, set two years after the events of the original game. It features a reprisal of the JENOVA theme and the return of the Reno.
Dirge of Cerberus is an "gun-RPG" game starring Vincent, a secret character who likes to pretend he's not a vampire. It takes place after Advent Children. Crisis Core is another prequel, this time for the PSP. Supposedly it's something for the PSP.
Eternal Crisis is just a rumor at this point, but it could be some remake of the original game. Crisis on Infinite Earths was an event ran by DC comics in 1985 in order to simplify their convoluted continuity problems. It is what Final Fantasy VII will be needing soon.
There are a few funky capitalizations unique to FFVII. Shin-Ra is the name of the evil power conglomerate, JENOVA is the power they mistakenly tried to harness in their SOLDIER genetic engineering program, which the TURKS recruit for. AVALANCHE is the force trying to stand up to them. I don't use the capitals all the time, but that's because I'm lazy.