According to FIDE's Laws of Chess, there are just two variants: Standard Chess and Chess 960, the latter being a variant where the starting positions are, within certain limits, randomized to make the game more about on the spot tactical thinking and less about memorizing prepared and proven opening lines.
Both variants are further subdivided by time controls - Classical, Rapid and Blitz - but those do not change the game as such, only the way certain tournament rules are enforced.
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