NOTE: I now document my favorite iOS apps for studying Japanese in the directory: http://www.mondaiji.com/directory/entries/category/view/64-japanese-study
Brak Software, Inc.
The author of this app is a saint. He has an uncanny gift for explaining the language in a transparent and comprehensible way, and has graced us learners of Japanese with this diamond of an app. Points that I've struggled with for years and years have been melting away in flames of coherence. "OMG! So that's how that works!" I wish I had this app when I first started studying Japanese. It would've rescued me from countless nights crying myself to sleep.
* Free lite versions available
* 2 levels available (beginner & intermediate). My fingers are crossed for an advanced level!
* It's an interactive textbook, so it stands out from all those other apps that flashcard the dictionary at you.
* Explains the framework of the language, building a solid foundation
* A bit on the expensive side of the app world, but well worth it.
Tae Kim's Guide to Learning Japanese
This is another on a short list of apps that works more like a textbook. In a similar vein to "Human Japanese," it leads through concise explanations of the language's structure and fundamentals so that you'll have a sound framework to build upon.
* It's free. What more could you want?
* Lots of examples to illustrate the text
* Comprehensive--goes up to a pretty advanced level
This straightforward yet colorful game makes my brief taking-the-subway-home-from-work kanji study sessions more enjoyable than mundane kanji flashcard apps. I have to admit--it's also a favorite app while doing my business in the outhouse.
* free version available
* It remembers words you're struggling with and challenges you with them.
* I wish it better avoided choosing uncommon words.