Puyo Puyo (or Puyo Pop outside of Japan) is a series of puzzle games. Puyo Puyo was originally a spinoff of the Madou Monogatari series, but eclipsed its home series in popularity. The first games were developed by Compile; Sega published the arcade versions and ports released on their platforms, while Banpresto published the SNES ports and spinoffs.
During a 1998 restructure of Compile, Sega gained the character rights to the series. Sega would receive the full rights to the series during Compile's collapse; every entry released after Puyo Puyo Box was developed by Sonic Team.
In Puyo Puyo, pairs of gelatinous "Puyo" are dropped into each player's 6x12 field. Connecting four Puyo of the same color together (they can be connected horizontally or vertically and do not have to be in a straight line) will cause them to "pop." As Puyo are eliminated, gray "Nuisance" Puyo are sent to the top of the opponent's field and will drop after the opponent has finished positioning their currently-controlled pair of Puyo. Nuisance Puyo can only be eliminated by clearing a set of adjacent color Puyo.
A player loses whenever their third column from the left is filled. In Puyo Pop Fever, Fever 2, and 7, filling the fourth column from the left will also result in defeat.
Compile (Madou Era)
The very first game in the series, released for the Famicom Disk System and MSX, was a simple single-player puzzle game with an optional competitive mode. The much more famous arcade game of the same name exclusively emphasizes the competitive aspects that the series would later be known for; this version was ported to the SNES, Genesis, and Game Boy, among other platforms. Sega had the Genesis and Game Gear ports reskinned with characters from the Adventures of Sonic the Hedgehog cartoon and released the resulting game, Dr. Robotnik's Mean Bean Machine, in the US and Europe. Nintendo followed suit with Kirby's Avalanche, which replaced the Madou Monogatari cast with Kirby characters.
Puyo Puyo's sequel, Puyo Puyo Tsu, added a simple mechanic that would drastically alter the way that the game was played: the ability to reduce or eliminate waiting Nuisance Puyo. Tsu's Compile-developed sequels, Puyo Puyo SUN and Puyo Puyo~n, add new (one-off) gimmicks to the "Tsu Rule"; Sun Puyo that grant bonuses in the former and character-specific powers in the latter.
Sega (Fever Era)
Puyo Pop Fever was the first main series game to be developed by Sonic Team. The new Fever Mode, activated by offsetting Nuisance Puyo before it can be dropped onto the field, generates several nearly-complete chains that can be used to turn the tide of battle. Additionally, the entire cast of characters, save Madou Monogatari protagonist Arle Nadja and series mascot Carbuncle, is replaced by newcomers. The game received a direct sequel that adds new characters but doesn't change Fever's core mechanics. The first Puyo Pop Fever is notable for being the final first-party Dreamcast game.
Puyo Puyo 7, which is currently the latest "main" Puyo Puyo game, introduces Transformations. Depending on the player's choice at the beginning of a match, filling the Transformation Gauge will either cause giant Puyo, (which are easier to match up) or mini Puyo (which drop in preset chains like Fever Mode) to enter the field for a short while. The game's roster of characters consists of a handful of new characters, along with a mix of Madou Monogatari and Puyo Pop Fever characters.
In addition, two Anniversary games were released: 15th Anniversary (before Puyo Puyo 7), and 20th Anniversary (after 7). These games feature a mix of characters from each Puyo Puyo "era" and, in addition to featuring the rulesets of the original Puyo Puyo, Tsu, SUN, and Fever, offer several different gimmick modes and four-player capability.
Out of the more than two dozen Puyo Puyo games, only four have (definitely) seen an official release outside of Japan. Puyo Puyo Tsu was released twice in the US and Europe: first as the Neo Geo Pocket/Color Puyo Pop, and later (untranslated) as Puyo Puyo 2 on the Wii Virtual Console. Minna de Puyo Puyo, a Game Boy Advance spinoff that was Sonic Team's first Puyo Puyo game, was localized as the GBA Puyo Pop. The N-Gage Puyo Pop was created specifically for the Western market. Finally, many versions of Puyo Pop Fever were translated and released internationally.
There are at least two English versions of the original (arcade-based) Puyo Puyo. The first is a directly-translated version of the System C2 game; it was released in Europe. However, there is little to no official proof indicating that Compile and/or Sega localized the title, which has led to claims that this release is an unlicensed bootleg. The second English version is Puzlow Kids, which is accessed by playing the Game Gear port of Puyo Puyo in an American or European handheld. According to the "All About Puyo Puyo Tsu" guide book, Compile was also planning a direct translation of Super Puyo Puyo before they were approached by Nintendo to make a Kirby-based game.