When the Xbox One launches in November, Microsoft's achievements system will undergo its biggest overhaul since its implementation. There'll be two kinds of achievements. Stuff you can earn that aren't achievements. Achievements for watching TV. And more.
First things first: you can now earn more than just regular achievements. Those are staying as they are (well, mostly, more on that later). Alongside them, though, games will also set you "challenges", which will be limited in availability, mostly pegged to real-world timeframes. Interestingly, these can span games, potentially including genres, series or a publishers catalogue.
Challenges can also be achieved cumulatively, an example given by Microsoft being "a headshot weekend challenge that requires players to cumulatively headshot 1 million baddies in a 3 day period. And every person who participates and meets the challenge’s goals gets the unlock on his or her achievement history and reaps its reward."
Both achievements and challenges can be "unlocked", both will display in your game history as having been something you've earned, but challenges won't count towards your gamerscore. I'd imagine they'll be used as much for older games as new ones, something like "revisit this old game's multiplayer for new challenges!".
Microsoft is unshackling achievements from games, and putting them in the "cloud". What does this mean? It means achievements for games can be fluid, added after a game's release (presently, new achievements can only be added via DLC), altered by feedback or even designed by fans.
Achievements no longer only unlock...achievements. Or avatar gear. They can now also be rigged to provide users with "digital artwork, new maps, unlockable characters, and temporary stat boosts".
This sounds great, but be warned, it's coming with a caveat.
Games aren't the only things that will grant you achievements. You can now earn them doing other stuff on your Xbox One, like watching video or using music apps. Completionists can expect to spend a lot of time...doing stuff that's not gaming.
On the bright side, only games will give you points towards your gamerscore. These extra-curricular ones are only for side-benefits.
That's all the main stuff. If you want to read the fine print, much of it useless without a good look at the systems actually in action, there's a rundown over on Major Nelson's site.