The Xbox One isn't going to do things the same way that the Xbox 360 did. Well, some things are staying the same, but not everything!
Mainly achievements are being overhauled and added to - oh, and Microsoft is adding something called 'Challenges'. You may be wondering what I'm talking about...So, I broke it up into bullet points below!
Your Gamerscore from your Xbox 360 will carry over to the Xbox One (YES!)
Achievements are not limited to games, but only games can add to your Gamerscore
The Xbox One dashboard will showcase your achievement gallery
You can see how far you've progressed in a game's achievement even before you start up the game (similar to Steam)
The dashboard will show you achievements that your friends have unlocked
Only available for a certain amount of time
Do not give out Gamerscore
Can cross over multiple games
Can be unlocked by community - everyone that participates in it gets credit.
save to your achievement history;
each have an icon
They can sometimes use Game DVR to capture what you've done
Devs can release more after the game's initial release.