Or better phrased, who has authority to do what? In the origin of our hobby the Game Master had both the responsibility and authority to govern many aspects of the game. How challenging is that task to accomplish? How friendly is a random cobbler on the street? What happens when you do pull the sword from the stone?
Many game systems as well as styles of play have shifted that mode and distributed some or all of the authority around the table. Here are a guidelines to use when classifying the player contribution in your game. Note, most of these terms are not mutually exclusive, there are many games that will fall into more than one category.
Your game will ask many questions of the players and give them space to shape the world their characters occupy. Their collaborations could range from small color details to the entire direction of the game.
Examples of system that encourage collaborative play: Apocalypse World, Fate Core, and Prime Time Adventures.
Something of a misnomer, GM-less games are actually GM-full! Everyone is working together to create the challenges and turn the character's choices into a narrative. GM-less games often assign certain roles such as "facilitator" to one player, but everyone's contributions are weighted equally.
Examples of GM-less systems: Fiasco, Microscope, and Fall of Magic.
Play to find out
Less about authority and more about the direction of the story, play to find out games look to the action of the players, and the consequences of those actions to determine the next event, challenge, or scene in a game. Play to find out games are often called "sandbox" as well. An open world where the action follows wherever the players go.
Examples of play to find out game systems: Burning Wheel, Blades in the Dark, and My Life With Master.
Alternately called PVP, games that feature player antagonism allow or encourage players to pit their characters against each other either as a single element or as the focus of the game. Games featuring PVP need special attention to assure that the actions of our characters are not conflated with the actions of the players. While my barbarian may be poking your paladin in the chest trying to start a fight, I certainly do not want to harm or antagonize anyone at the table.
Examples of games that feature player antagonism: Paranoia, Monsterhearts, and Carolina Death Crawl.
Similar to GM-less play, games with rotating authority put different people in the role of GM at different times, either to adjudicate a particular game session, scene or element of play.
Examples of games with rotating authority: Polaris, Archipelago, and Night Witches.
A classic style of play where the GM has an adventure prepared and the players will navigate their characters through the story as they solve mysteries, evade traps, and struggle against foes the GM has planned for in advance.
Examples of games that encourage strong storylines: Pathfinder, Unknown Armies, and Werewolf: The Forsaken.