Emergent Story

and this Old School Gamer
"Th[e] story is an emergent phenomenon resulting from the action and reactions of the player characters as they interact with each other and the dynamic world in which they live." (taken from Dresden Files Accelerated)
While I hesitate to expound on a single way to have fun playing the games that together make up our hobby today, I do like to talk about my preferences and how I like my own games to be. Dresden Files Accelerated is a Fate Accelerated game, meaning it uses the Fate system derived from the Fudge narrative engine (using Fudge Dice - six sides, two minus, two blank, two plus) and streamlined under the Fate Accelerated version. It is published by the fine folks at Evil Hat Productions who also do the Dresden Files in three big volumes and the Fate Core and Fate Accelerated rule system books. They are all high quality products which fall under the indie game umbrella as I understand it.
Games which emphasize narrative, or "story", sometimes feature mechanics to distribute control of story among the players. Players are empowered through use of various in-game rewards, to spend the game's currency to interject story elements which may occur to them as they play, things which can be beneficial to their character or just something the player thinks is entertaining. Ultimately the referee, who adjudicates the rules and has a better understanding of the setting than any other player, has control of what is allowed, but many such narrative game systems make that control a "last resort" and discourage its use.
As mentioned, Fate, on which Dresden Files and Dresden Files Accelerated is based, uses Fate points or tokens as the currency that can be spent to inject narrative elements into the game during play sessions. This happens in a couple of ways. The referee may offer the player a Fate point if some event can directly affect how the character reacts based on certain PC characteristics called Approaches. In turn the player my reject the suggestion, paying a Fate point to "resist" or they may go with the "compel" and bank the Fate point to be used later to give their character a boost (such as re-rolling the dice) or adding some narrative element of their own invention into play.
Fate points are a relatively soft way to add player control to the game's narrative. Although there are other systems which give the players greater control of story elements, all such mechanics to a degree alter the dynamics between players and referee and together with games that use complex character builds and extensive amounts of rules that attempt to cover everything so that referee rulings are virtually eliminated, all seem to be universally directed at mitigating the negative effects a bad referee may have on the game despite often being labeled "player agency".
Before I totally wander off-topic, let me return to what emergent story means to me. Story is what the quote above states - it is what happens during the game. Story at my table is not written by the referee to be acted out by the players in some form of tabletop play production.  It is also not something the players (or referee) dictates as it occurs to them at the table in some "oh wouldn't it be wonderful if this happens now" wish fulfillment. The choices and ideas the players make in response to the setting as described by the referee and the results of the dice are what creates the story. It is cooperative, but not necessarily collaborative. Depending on the choices made and the outcome of the dice, the story may go very badly for some or even all of the players. The "high stakes" of putting the PCs into "danger" adds immeasurably to the excitement and pay-off of play using the old school style and letting the dice decide. As referee or player, I may roll Fudge Dice, but I roll in the open and I don't "fudge" dice!