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
The Best Apps
If you plan to study Japanese (or any foreign language for that matter), you have loads of new words to learn. You're gonna need a solid flashcard program. Sure, there are Japanese-specific study programs that have volumes of included flashcards, but there's a significant memorization advantage to making your own customized flashcards. Namely, the words you'll study are relevant to your current Japanese study--they're words you're working on and/or covering in class.
I tried a bunch of highly-rated flashcard app demos before settling on Flashcards Deluxe because it was the only one I found with a critical feature I was seeking. It allows you to create cards with up to 5 "sides." So I put the kanji on side 1, furigana on side 2, meaning on side 3, and a mnemonic hint or note on side 4 (if I can come up with one). This feature alone sold me on Flashcards Deluxe, but awesome also is the "spaced repetition" study algorithm and ability to import cards via the developer's website.
Flashcards Deluxe is now my go-to app for studying Japanese vocabulary.