"Powered by the Apocalypse" isn't the name of a kind of game, set of game elements, or even the core design thrust of a coherent movement. (Ha! This last, the least so.) Its use in a game's trade dress signifies ONLY that the game was inspired by Apocalypse World in a way that the designer considers significant, and that it follows our policy wrt others' use of our creative work.
The Apocalypse World engine was first used in Apocalypse World, published by D. Vincent Baker in 2011. The game's flexibility, as well as Vincent's open encouragement to create custom moves and content, has led to a number of hacks and games based on the Apocalypse World engine.
Many other games have been published using the "Powered by the Apocalypse" logo. As noted in "An open letter re: Powered by the Apocalypse", D. Vincent and Meg Baker do not consider "Powered by the Apocalypse" to represent any particular system. However, in practical terms the games that use this logo share many common features; the use of "moves" as packaged pieces of rules text, a list of defined principles and an agenda for play, and many a common dice framework using two six-sided dice.
D. Vincent Baker maintains a forum dedicated to talking about Apocalypse World and all of the hacks and designed based on the Apocalypse World engine. See the weblink below for details.
Powered by the Apocalypse games are all centered around resolving what characters do as Moves. All characters have access to a default selection of moves focused on the main subjects of the game. For instance, there is a default attack move in Dungeon World called hack and slash as this is core of the dungeoneering experience, but in Apocalypse World you can only find a move like seize by force as the game focuses on what one would extract from others in a world made of scarcities.
Apocalypse World, Dungeon World, and most other PbtA games are class based with the class selected for the given character giving them access to a number of moves beyond the default. These moves can allow them abilities above and beyond the normal, like the Hypnosis ability of the Monsterhearts Vampire, can give them additional resources, such as the Apocalypse World Driver's "My other car is a Tank," or may simply make them better at moves everyone has access to, like the Apocalypse World Gunlugger's "Insano like Drano."
Some moves resolve automatically, but most involve an element of randomness. A player whose character makes a move rolls two six-sided dice and adds the relevant modifier, which varies by move and game. A result of 10+ is a total success, and the character achieves their goal. On a result of 7-9, the character achieves a partial success, in which they mostly get what they want but also face some consequences or have to make do with a lesser version of their goal. A roll of 6 or less results in a "miss," and the Master of Ceremonies or Game Master makes a move of their own, with negative consequences for the acting character.