How many styles are there of playing chess

That depends on how you look at it. One set of styles is positionalas opposed to combinative. The positional player tries to get the pieces on the board in specific formations or positions that he or she is familiar with and knows how to play to his or her advantage. The combinative style is one where the player uses his or her pieces in various ways to attack the other player's pieces and gain a checkmate. Another set of styles can be looked at as where a player uses his pieces to take control of the center of the board by occupying the center. On the other side of that coin is what is called the "hypermodern" style, which is where the player takes control of the center of the board, not be occupying it but by attacking the center from the edges. Both styles seek to control the center of the board but in different ways.