Nengō Dentaku 年号電卓 (iOS)
This app gives the reign/era years next to Western years in a simple, convenient format going back to 神武天皇, 660 BCE. It does not provide the beginnings of lunar months or zodiac signs corresponding to the years. Free.
Nenrei/Eto/Nengō/Kōki/Seireki Dentaku 年齢・干支・年号・皇紀・西暦電卓 (Android)
Converts Western years to era, emperor, and zodiac names back to the year 660 BC; you can also browse the eras and emperors.
All国語辞書, Japanese ⇔ Japanese (Android, iOS)
This dictionary searches across a variety of online 国語辞典, including Yahoo!, Weblio, Goo, Excite, Kotobank, Google, and more. It supports handwriting recognition for kanji and kana on Android. Free.
This searches Goo's site for 国語辞典 entries on various topics. It simply displays information from Goo's site and does not support handwriting recognition. Free.
Weblio 辞書 (Android)
Weblio 辞書 searchces Weblio's site for encyclopedia and dictionary entries. Free.
Japanese Verbs (Android)
This app provides a verb conjugation dictionary (including polite and plain forms) as well as quizzes. You can search in either Japanese or romaji.
Japanese Text Analyzer (Android)
This is a morphological analysis tool for Japanese language, providing linguistic information, proofreading aid, dependency analysis, ruby glossing, kana to kanji conversion, and WWWJDIC lookup. Free.