There are a lot of really dumb apps in the Apple app store but if you do a bit of hunting around there are also some great ones to be found. Here's a few I've found particularly helpful for bible study/theological learning. I tried these out on an iPod touch but they should work just as well on an iPhone. Some of these apps also have an iPad edition available.
ESV bible (free)
While this one has less features than some other bible apps, it is well designed and easy to use. It features the popular English Standard Version of the bible. It also does not require internet access for any of the features which is great if you are an iPod touch user and not always within range of a WiFi hotspot
YouVersion seems to be one of the most popular bible apps. It includes many English translations and some foreign language translations. Some of the available translations can be downloaded for offline use, some you need an internet connection. Other features include reading plans, bible commentary from other users and being able to search for nearby church services and events.
Legonier Ministries (free)
The Legonier ministries app includes daily devotions and messages from R.C Sproul. It also includes an extensive library of sermons and articles on a wide variety of topics by teachers such as Jerry Bridges, D.A Carson, Kevin DeYoung, Joshua Harris, John MacArthur, Albert Mohler, John Piper and many more.
Mars Hill (free)
The Mars Hill app features an extensive library of sermons from popular preacher Mark Driscoll. Sermons are available in audio and video formats. The app also includes the Mars Hill Church blog and a lot of original worship music from the church.
GoodReader (US$1, AU$1.19. free but limited version also available)
Not a Christian app, but it definitely has uses for Christian content. There are heaps of great free Christian ebooks online in .pdf format from sites like desiringgod.org or gfa.org. GoodReader makes it really easy to read these or other publications in pdf or Microsoft office formats on your iPod touch or iPhone. There is heaps of options for transferring books to your gadget including email, FTP, downloading straight from the web, google docs and WiFi (some of these options do cost an extra dollar to unlock). When reading the books you can read them with the original document's formatting or reformatted for easy reading. The app will automatically open the document to where you were last reading.
Anything worthwhile I've missed?