See ceejayoz's answer for the various iPhone-specific stuff (icon, fullscreen mode), but in order to store the entire app locally (and run offline), you'll need to look at what's called a "cache manifest". This file, linked to in the opening html tag on your page, lists every resource the app needs to store locally.
Additionally, in order to store user data, if need be, you'll need to look into the client-side database. I don't know as much about that, so I won't try to explain it. :P
Apple has a decent page here: http://developer.apple.com/safari/library/documentation/iPhone/Conceptual/SafariJSDatabaseGuide/Introduction/Introduction.html#//apple_ref/doc/uid/TP40007256-CH1-SW1
that talks about both the cache manifest and local database storage. This should help explain what you need to do in order to make your application run offline.