So, I've run a short Apocalypse World campaign (set in the ruins of the Hoover Dam) and have an occasional Monster of the Week game, and I thought I'd check out Dungeon World. This is the same game design, but with the trappings of Dungeons & Dragons. As a long-time D&D DM (I run the D&D Encounters program at Maplewood Hobby) I've had a great deal of experience running all different flavors of the game.
Basically the idea here, is that DW is a story game. That means it's focused on narrative over dice rolls, though there's plenty of rolling dice. It's also focused on the fiction over the game mechanics, though there's plenty of game mechanics. Essentially, on the Narrativist-Gamist-Simulationist axis, it's about three-quarters Narrativist/one-quarter Gamist.
It's worth mentioning that Vince Baker, author of Apocalypse World, also did Dogs in the Vineyard. If DitV is Mr. Baker's meditation on dice mechanics in role-playing games, AW is his meditation on character sheets. When you choose a character type, you pick up that "playbook," which is a character sheet with all the character-specific options already printed on it. As you develop, you select from the other options in the playbook. Some of them give you new things you can do, while others just improve your ability to do things everyone else can do.
This fits very well with the D&D concept of "character classes." Character classes in D&D have traditionally been niches that defined what your character was capable of, but gave some room for expansion; it was a fairly uniform approach, so that every instance of the same character class was fairly similar. As time went by, people developed tools to allow people to customize the classes to better fit their vision of what their character was. Originally, there was some playing around with the experience point structure of white-box D&D by Gary Gygax, where you could add abilities to your character class but would need to earn more XP to advance. Then there was the kit system of 2nd Edition AD&D, which essentially codified the earlier system. 3rd Edition D&D strongly resembled RoleMaster, in that a character class just defined what you found easiest to learn, without really preventing you from doing anything else.
In AW-style gsmes there's enough structure to feel like D&D, but there's also possibilities for swapping in other abilities. They support niche protection -- there is only one of each playbook at the table -- but still let you gain some other abilities as well. It's a pretty good fit, more abstract than the original system, and is simpler to boot.
One of the big draws for a cheapskate like me is the huge amount of quality free material online. Check out the Dungeon World Tavern for just a sampling of the kind of thing available. My thinking is that this will also work well with the One Page Dungeons I'm a big fan of -- the One Pagers are practically tailor-made for DW, as the system encourages limited DM pre-planning.