When the question is asked how many ways to play D&D, you will usually get an exhaustive list of play styles like Hack and Slash, Monty Haul, Exploration, etc... but these are more thematic styles than actual styles of play. When someone asks me the same question I say that there are two true styles of play, and that things like Powergaming and Roleplaying are just preferences within one of the two play styles. The two styles are DM Driven games and Player Driven Games.
DM Driven Games are what most people are used to. They have a DM that prepares an adventure or perhaps runs a pre-made adventure, that establishes the world for the players to interact with. If the first person to speak in game is the DM then you are probably playing a DM driven game. In a DM driven game, the DM usually presents the scene to the players, who then react to the DM's description.
Example of Play:
DM: You are in a long corrider made of hewn stone. 30 feet ahead stands a large Oak Door. The walls are covered with a purple mold which glows dimly.
Player: I open the door.
DM: The door opens revealing a large circular room with a fountain in the center. Along the walls are statues of armored men and women carrying spears and shields.
Player: I drink some water
DM: The water tastes of almonds, suddenly your whole body begins to feel hot. As your throat swells up you struggle to breath. Gasping for air you lie on the floor like a caught fish, powerless as the poison takes effect.
Player Driven Games are the opposite of DM driven games. Instead of having the DM provide the situation to the players, the players provide situations for the DM. Many people do this to an extent, usually when the players are in town and are getting supplies and such.
Example of Play:
Player: Where am I?
DM: You are sitting in a tavern.
Player: I look around what do I see
DM: You see several customers at various tables nursing their drinks and avoiding eye contact with each other.
Player: I stand up on the table with a tankard raised and shout "Bottoms up me fellow travelers, the Ale is on me!"
DM: Everyone applauds you and soon the tavern is awash in merry drinking songs.