It's just a Twitter clone, that charges users for membership, instead of monetizing them through ads or selling their data. A lot of people think that will lead to a better service, so it got a lot of hype recently.
The twitter clone is a proof of concept. A better description would be that its a paid social graph designed to be used by third party apps. Just like Facebook apps or apps that use twitter to sign on, app.net can be used as that identity- the difference being that app.net is promising not to put arbitrary restrictions on how it is used.
Hmm, that's disappointing, especially after the initial post that was about Twitter's potential to become the web's real-time feed API.