In fighting games, the same character can look different simply by changing around his or her costume (and in recent 3D games, this is the norm). In 2D games, however, this would entail too much graphics work, so instead, the color palettes on the character’s artwork are changed.
The first instance of character-specific color changing was in Street Fighter II′: Champion Edition, which had two color schemes per character (it was also the first Street Fighter game that allowed both players to choose the same character, hence the need).
Ever since, most fighting games have allowed players to pick one of a few color schemes simply with the buttons they use to choose the character. Some players even have favorite schemes.
In fact, some recent games have a large selection of schemes (notably Melty Blood, which has up to 36 different ones in the arcade version of Act Cadenza ver. B).
However, sometimes these colors blend into the background (good defaults obviously should not), giving the player an unfair advantage in some cases.
In home-use versions of games, there is often some system allowing the player to change a character’s colors. This doesn’t stop at that, though; sometimes even the colors of moves’ graphical effects can be changed.
This allows a flair of customization to a game that can actually be used in combat (perhaps even network battles depending on the game), so it is popular with some players. However, because most 2D fighting games use a limited palette for character colors such that each color defines a specific region of a character’s appearance, the unaccustomed often have a hard time making heads or tails of how to customize with this feature.