Many accounts of some "narrative games" (your term, not mine) have been given in this thread. I've posted a lot about Burning Wheel. I have posted a bit about Apocalypse World and Dungeon World, and so have others (eg @AbdulAlhazred, @Campbell).
BW, AW and DW all share the same basic authority structure as D&D - the GM describes imaginary situations, in which the PCs are present, to the players; the players say what their PCs do; as a result of the PCs doing things, the situation changes or develops in some fashion.
The key technical differences are in the heuristics, principle and rules that guide the participants in what sorts of situations to describe and what sorts of things happen as a result of the PCs' actions. I've explained this in relation to Burning Wheel at length upthread, including in reply to you. Here is it for Apocalypse World (Dungeon World is pretty similar):
*If the players are looking to the GM to see what happens next, the GM says something about what is going on in the fiction. The rulebook calls this the GM making a move; and it provides a list of generally-described moves for the GM to make, the most important feature of which is that "nothing happens" is not on the list.
*When the GM makes a move, it is by default a "soft" move, in the sense of establishing some threat or risk or opportunity or promise, that is pending but not yet realised. This obviously invites the players to have their PCs do something in response.
*When the players have their PCs do something, by default the GM says what happens as a result - either another soft move, or if the PCs' actions don't respond to an earlier soft move, then a "hard" move as the GM brings home the prior threat, risk, opportunity, promise etc.
*However, as an exception to the above, certain player action declarations trigger defined player-side moves (eg if a player has their PC read a charged situation, this triggers the move Read a Sitch). These moves are resolved by a dice roll; each is a little sub-system. (There is a fairly close resemblance here to the various sub-systems for roll resolution in Classic Traveller.) Generally, if the result is 7+ the sub-system rules set out who gets to say what happens next, and what sorts of parameters constrain what they say. (Think of this like reducing a character/creature to zero hp in D&D - this can be narrated as death or as unconsciousness, depending a bit on further features of the situation and a bit on GM or player choice.) If the result is 6 or less then generally the GM is permitted to make a hard move. (I say "generally" because some player-side moves are more specific about what happens on a 6 or less).
*When the GM is making their moves, in addition to the technical structures I've just described, there are a variety of over-arching principles that also apply. These serve a few different purposes - to establish verisimilitude; to keep the focus of the action on the PCs; to encourage the players to bring their PCs thoughts, hopes and "inner lives" to life; and to reinforce that "nothing happens" is not an available GM move.
*The rules also tell the GM how to undertake and structure their prep, with the overarching purpose of prep being to give the GM interesting stuff to say when everyone looks to them and so they have to make a move.
*When the GM makes a move, it is by default a "soft" move, in the sense of establishing some threat or risk or opportunity or promise, that is pending but not yet realised. This obviously invites the players to have their PCs do something in response.
*When the players have their PCs do something, by default the GM says what happens as a result - either another soft move, or if the PCs' actions don't respond to an earlier soft move, then a "hard" move as the GM brings home the prior threat, risk, opportunity, promise etc.
*However, as an exception to the above, certain player action declarations trigger defined player-side moves (eg if a player has their PC read a charged situation, this triggers the move Read a Sitch). These moves are resolved by a dice roll; each is a little sub-system. (There is a fairly close resemblance here to the various sub-systems for roll resolution in Classic Traveller.) Generally, if the result is 7+ the sub-system rules set out who gets to say what happens next, and what sorts of parameters constrain what they say. (Think of this like reducing a character/creature to zero hp in D&D - this can be narrated as death or as unconsciousness, depending a bit on further features of the situation and a bit on GM or player choice.) If the result is 6 or less then generally the GM is permitted to make a hard move. (I say "generally" because some player-side moves are more specific about what happens on a 6 or less).
*When the GM is making their moves, in addition to the technical structures I've just described, there are a variety of over-arching principles that also apply. These serve a few different purposes - to establish verisimilitude; to keep the focus of the action on the PCs; to encourage the players to bring their PCs thoughts, hopes and "inner lives" to life; and to reinforce that "nothing happens" is not an available GM move.
*The rules also tell the GM how to undertake and structure their prep, with the overarching purpose of prep being to give the GM interesting stuff to say when everyone looks to them and so they have to make a move.
That's a description of some key technical features of AW. Something that @Campbell has been emphasising is its non-technical differences from some fairly typical D&D play. The GM's principal goal, in presenting situations and narrating consequences, is not to reveal a world or setting or environment to the players. It is, rather, to present risks, threats, opportunities etc that will prompt the players to declare actions for their PCs based on their sense of what their PCs want and feel and hope for when confronted by these risks, threats, opportunities etc.
And the principal goal of a player in AW is not to (i) learn details of the GM' setting so that (ii) they can explore it, or manipulate it, so as to achieve some goal - often wealth and status - for their PC. The player's goal is to be their character, who has personal and often rather intimate hopes and concerns that the circumstances of the apocalypse - both material and social - place under pressure.
I don't think we've had any AW actual play posted in this thread, but I've posted plenty about Alicia and Aedhros and Thoth from Burning Wheel play. If you look at those accounts, you might notice that there is no "adventuring", in the sense of going somewhere strange or unknown on a hunt for loot or a quest to do something-or-other. Aedhros and Alicia find themselves in the underbelly of Hardby, a port town, and rob a tavernkeeper and then a port official's rooms. Then they are taken by Thoth, who is trying to bring the dead back to life as undead; Aedhros helps collect bodies for him, while hoping but failing to reconnect in some fashion with his Elvish heritage.
As I think @Campbell posted not too far upthread, a lot of what some posters in this thread have characterised as "downtime" is, in a game like AW, the focus of play.
