Final Fantasy

Final Fantasy is a popular series of RPGs produced by Square (aka Squaresoft). Most Final Fantasy games have a surprising level of detail given to the plot and character development. All the games in the series occur in different universes, and are unrelated, except for the occasional cameo (such as Cloud's apperance in FF Tactics). However, there is a tradition where many of the games have characters named (Biggs/Vicks), Wedge and Cid, as well as recurring races, such as the Moogles and Chocobos. The battles in these games are usually semi-turn based, using a system known as the Active Time Battle - introduced in Final Fantasy 4. The battle system differs somewhat among all the games, particularly in the use of magic. As of this writing, there are a very large number of games in the series, including the following (note that the information about the plot is only the introduction, and is by no means the entire depth of the story line -- in fact, in many cases the introduction belies the fact that the main story is almost completely unrelated to the introduction, which is often used simply to introduce the characters.)
Final Fantasy 1, 2, and 3 (for the Nintendo Entertainment System and Nintendo Famicom). Only the first in the series was available in America.
Final Fantasy 4 (for the Super Nintendo Entertainment System). FF4's plot focuses on a main character named Cecil. The game begins when Cecil and the air force he captains (called the Red Wings) are forced to steal a Crystal. Cecil objects about this to his king, who promptly demotes him and sends him off (along with his friend Kain) on an errand to carry a package to a place called Valley Mist. This game was released as Final Fantasy 2 by Square of America. There is also an unoffical version of this game called FF4 Hard, which was translated into English from Japanese by players, as opposed to the standard version, which was translated into English by Square. 
Final Fantasy 5 (for the Nintendo Super Famicom). This game was not available in America.
Final Fantasy 6 (for the Super Nintendo Entertainment System and Nintendo Super Famicom). FF6's plot focuses on a character named Terra. The game begins with two of the Empire's henchman, named (Biggs/Vicks) and Wedge, and Terra, who's mind is being controlled by a device called the Slave Crown. They attempt to take a frozen creature called an Esper from a northern town, and in the attempt, Terra is knocked unconscious, and later saved by a member of the underground rebellion against the Empire. No relation to the Empire in Star Wars. This game was released as Final Fantasy 3 by Square of America.
Final Fantasy 7 (for the Sony PlayStation and PC). This is the first Final Fantasy game in which the characters had both first and last names. It is also the first game where the character's names aren't arbitrarily capitalized. FF7 is also the first 3D Final Fantasy. FF7's plot focuses on a character named Cloud Strife. Cloud is beginning his employment for a group called AVALANCHE, headed by Barret Wallace. They are attempting to sabotage a Mako reactor, a device which drains energy from the Planet to generate electricity, to create monsters, and to create Materia, magical orbs. These reactors are created and maintained by a corporation called Shinra. 
Final Fantasy 8 (for the Sony PlayStation and PC). This game featured a huge leap in graphics and cutscene quality, but many were unhappy with the cardboard characters and uninteresting plot.
Final Fantasy 9 (for the Sony PlayStation and PC). A return to Final Fantasy's roots - likable characters, a main character with an unknown past, and stereotypical examples of the original series' various character classes. Very enjoyable and fun.
Final Fantasy Anthology (for the Sony PlayStation). This game is a compilation of FF5 and FF6, including some CG movies not available in the original games.
Final Fantasy Chronicles (for the Sony PlayStation). This was the second multi-release of old games for the PSX, including FF4 (which was included in the Japanese release of FF Anthology, but not in the US release, but was previously released in the US as Final Fantas 2 for the Super Nintendo) and Chrono Trigger (also previously released for the Super Nintendo). FF4 featured new CG animation, while Chrono Trigger featured new anime style cutscenes.
Final Fantasy Tactics (for the Sony PlayStation). This game uses a completely different battle system then the other games, comparable to a game of chess in many ways. This game was Square's answer to Atlus' successful Ogre Battle series, which is similar both in concept as well as plot to FF Tactics. In stark contrast to the other PSX Final Fantasy titles, Final Fantasy Tactics used a 3d, isometric, rotatable playing field, with bitmap sprite characters (the exact opposite approach taken with Final Fantasy 7). The plot revolves around Ramza Beoulve, the youngest son of an aristocratic nobleman, and his best friend Delita, who was taken in at a very young age by Ramza's family. Ramza and Delita are caught in the turmoil of the Lion War, a power struggle between two rival princes for control of the kingdom. Ramza must choose between his concience and his duty, while Delita must come to grips with his family's lack of nobility, despite his upbringing by the Beoulves.
There were also Final Fantasy Legend and Final Fantasy Adventure games for the Nintendo Game Boy), but properly speaking these were part of different series, the Secret of Mana and something-or-other.
There was also Final Fantasy Mystic Quest for the SNES, which was rather different than the other games. The combat system was different as was the game's format. An odd little gem.