How the Achievements work will be specific to that individual game, at least in the case of New Super Mario Bros. U for Wii U. However Nintendo has confirmed that MiiVerse features are innate to the Wii U itself, and thus every game will feature MiiVerse features. That means its very plausible that this Achievement system I will detail here will be THE Achievement system built into all Wii U games.
The Wii U Achievement system works like this. While playing New Super Mario Bros. U, you can gain Achievements by completing certain conditions as you make your way through a level. Conditions explained include collecting all the Star Coins hidden on a level in one go, completing the level without being hit by a single enemy. If you manage one of these conditions, called a “Super Play” by the developers (tentative term I imagine), then you will be prompted to leave a message regarding your awesome performance, that will then be shared with other players online via the MiiVerse, as well as with your own friends who own the game. Nintendo was adamant that there are “A LOT” of different conditions, or Achievements, that players can strive for.
You can hear Nintendo discuss this in the E3 2012 Wii U Developer Discussion Video. Skip to around 6:30.
Messages from other players on the MiiVerse will then be spread throughout the actual game as you are playing, and revealed when you meet or fail to meet these Achievements; alerting you to the fact that these Achievements exist, or dropping hints as to what it may take to fulfill a Super Play.
It is easy to imagine this same Achievements system being applied to other games, in the same manner that it is applied to New Super Mario Bros. U. Will this be the bonafide Achievements system for the Wii U? Seems to me like it will be, but we’ll have to wait and see what else Nintendo reveals regarding an Achievements system for Wii U, and whether or not these games will have Badges or Emblems or Points or Trophies associated with earning these Super Play or Achievement conditions and how they are shared with your online friends and the world at large via MiiVerse.
You can get an idea of how this works in the screenshots above and below.