After a passport/identiy check it would technically just be a highly available service to cryptographically sign blobs, coupled with a mechanism to handle retractions of signed data in case of compromised keys and a decently advanced notification system for early detection of suspected fraud.
On a societal level however, this would enable anything from frictionless contract signing apps to building a Kickstarter for organizing ad-hoc grass-roots referendums whose results are provably meaningful.
All of the contractual agreements until then were signed digitally through DocuSign and they even have a nifty little iphone app. Of course they were all contracts written by my lawyers, but that counts as data. The concept of a written signature is far less relevant than it used to be. As people have mentioned actually signing a piece of paper digitally or otherwise isn't required for many legally binding contracts. (For example, my previous lease on my apartment was an email exchange between me and my landlord agreeing on the terms. Had he or I broken the terms of it, it would have been enforceable in court (in IL at least; anywhere else I couldn't say.))
- an OAuth(-like) API for other sites, e.g. Reddit, to sign micro-data on your behalf.
- the ability to retract signatures after time X and notify subscribers of that data of it.
- [optionally] a fraud detection algorithm to give advance warning of possible malicious usage (app compromised or your app password.)
- the ability to request confirmation of a signature before time X for cases where retraction after that moment is unaccaptable.
That would make it possible for webapps to have every interaction with it be provably performed by the user, turning it into a legally valid audit trail.
The other comment's DocuSign suggestion comes closer, but I also doubt DocuSign offers support for third parties to do micro-signing through an API.