Chocobos come in various different chocobo types.
Generally, if there is any difference in the abilities or disposition of a chocobo, it is based upon their color. Below are a list of traditional colors for chocobos, where they appear, and what that means in terms of their abilities, if anything.
Yellow chocobos are the most common of all chocobo colors and can be found in every game where chocobos appear. They are the most easily tameable of all types, and are usually the only color available for rental by the player.
Blue chocobo are a weaker chocobo type than black chocobos. They are able to cross rivers and streams, but typically cannot fly. When they participate in battle they are generally magic casters. Blue chocobos sometimes have a darker variant that can cross oceans.
Green chocobo is a weaker chocobo type. Their abilities are variable but in games where they participate in battle they act as healing and support characters.
Red chocobo act as offensively powerful chocobos, but are still much weaker than other breeds.
Black chocobos were one of the first varieties of chocobo to be seen, and are one of the most common. Though rarer than standard yellow chocobos, black chocobos are able to fly, though not with the same versatility as airships. They are common sights as enemies in games.
white chocobos are very rare, and are among the highest rank of chocobo breeds when they appear. Similarly to green chocobos, they support other characters when fought in battle, and out of battle in some games can restore the party's MP.
Gold chocobos are the rarest and most valued type of chocobo. They are valued for their ability to fly, supplanting black chocobos as the only breed to do this in later games. Acquiring a gold chocobo is often the result of a long and difficult sidequest.
Brown chocobos are also found in Final Fantasy Tactics A2: Grimoire of the Rift, where they can use the Choco Guard ability to cast Regen and increase defense. They also appear as enemies in Final Fantasy Tactics S.
Added in Final Fantasy XIII-2 the silver chocobo is a defense-oriented Sentinel. While, like many of the chocobos in Final Fantasy XIII-2, it's in the running for best Sentinel in the game, it really shines for being the earliest, and arguably best, racing chocobo in the game. It learns the best racing abilities easily and its stats hit the max for the racing game with some reasonably easy grinding for the point in the game it's found at.
In Final Fantasy XIII-2, Purple Chocobos are Synergists. Like many of the chocobos, it's one of the best monsters for its role. They are native to Bresha Ruins 100 AF and 300 AF. In Chocobo Stallion, they appear as one of many colored racing chocobos.
In Final Fantasy VII, the pink chocobo only appears in the races. Pink chocobos also appears in Final Fantasy XIV.
Fat Chocobo is a different type of chocobo, and never appears as a playable chocobo. Instead, the Fat Chocobo serves another role, by storing the player's items in the series' early games, appearing as a summon in Final Fantasy V, Final Fantasy VII and Final Fantasy VIII, serving a storyline role by being the king of all chocobos in Final Fantasy IX, and as a mount in Final Fantasy XIV.
Baby chocobos, known as "chicobos" in Final Fantasy VIII, appear in certain games of the series. In Final Fantasy V Boco and Koko have children, and in Final Fantasy VIII the player can obtain a chicobo from a Chocobo Forest to play as in the Chocobo World minigame.
A baby chocobo named Bobby Corwen is revealed to have hatched from an egg in the Black Mage Village in Final Fantasy IX. The tiniest baby chocobo type appears in Final Fantasy XIII, with Sazh's Chocobo chick being small enough to use his afro for a nest.
Final Fantasy XIII has two different types of chocobos, Pulse and Cocoon breeds. Chocobos roaming the Archylte Steppe in Gran Pulse live in the wild and are much larger than their domesticated Cocoon counterparts, as well as having a more vicious appearance. Cocoon chocobos can be found in Nautilus' petting zoo. Sazh's chocobo chick is a Cocoon chocobo.
Final Fantasy XIII-2 presented a mutant breed of Gran Pulse chocobo, known as the rambunctious chocobo, appears in certain areas that have been under the influence of the paradox for an extended period of time. They have different plumage and demeanor with red feathers with purple tail, and unlike normal chocobos, will continually eat from the player's supply of Gysahl Greens. Some rambunctious chocobos that aren't in fixed locations will actively avoid the player, and most will do so when dismounted regardless of whether or not the player has greens to feed them.