Here is a list of useful apps for learning Japanese. Suggestions for additional apps welcomed.
Anki – the queen of all spaced repetition flashcard apps. Free. Android, iOS
Human Japanese – an excellent eText available for a low price and with a free trial. As of this writing there is a beginner version and an intermediate version available. Android, iOS
imiwa? – a dictionary and more – speech, sentence analysis, kanji, kana, personal notes. Free. (Thanks nahcirn) iOS
Innovative – JapanesePod101.com app. The website charges a subscription fee. Android, iOS
Kanjigames – a suite of apps for studying Kanji and phrases in game form. I haven’t tried them, but, from the webpage, they look like they might be both fun and useful. Free but with in-app purchases. (Thanks Oliver) Android, iOS.
Kanji Study – an app that helps you study kanji meanings, usage and writing. Can be set to present kanji by grade level, by JLPT level or by custom groups. The lowest levels are free, and a one time fee gets you access to everything. Android, iOS
Line – Send free text messages and make free calls worldwide. Ad supported. The ads can be removed with a paid option. Android, iOS
Memrise – app for the website. Tons of lessons for learning kanji, kana, vocabulary, etc. Free. Android, iOS, Windows
Mindsnacks – app for playing vocabulary games. You get a few lessons for free but others can be purchased. iOS
Sticky Study – a spaced repetition flashcard app. There are special kana and kanji versions, but you can also make your own flashcards or import them from quizlet. Low cost. iOS
Takoboto – Japanese dictionary and language learning tool. Look up words by roumaji, kana or kanji. For each word you can get the translation, the reading, the meanings of the individual kanji, stroke order for the kanji and sample sentences. Free. Android, IOS
Tangoristo – Japanese reading for language learners. TangoRisto analyzes short Japanese texts from the web to facilitate your reading flow. This app encourages you to read and explore the vocabulary in short texts from the Web. You can review the vocabulary for each text, filter it by your reading level and bookmark your favorite articles. Free. Android, iOS. Thanks Yanock
TuneIn – app for the free website. Lets you listen radio stations in a variety of languages. No lessons. Android, iOS