I installed the app (it's free), and took a look at it -- unfortunately it's not very robust, as there's no way to see gear (unlike Omen of Clarity's iArmory web app), but it does run great and look shiny. Upon running the app, you can just punch in your character's info (or anyone else's -- all you need is a name and realm), and it'll get added to your list. Choosing one sends you to screens where you can see information about professions and talents, ability stats, or combat stats.
And that's pretty much it -- more is planned, including reputation, talents, and items panes. EU and China support should be up in a few days (right now, it only works on US characters, and of course it depends on the Armory being up to update). Very nice for a first try, but we still can't wait until we see something a little more in-depth, or, dare we ask whoever's working in Blizzard's mobile division, even official.
Note: This writeup is current as of version 1.0. Later versions of the app will likely include added functionality.
Update: Good news from OOC: he and Rudi are planning to work together on the Characters app, which means it should soon have all the functionality of the iArmory web application, and then some. Saved characters? Character compare? Guild progression and be.imba integration? Yes please!