Note that the majority of Japanese-English dictionaries available for free offline use the same dictionaries so it simply depends on which interface you prefer. Most of the offline dictionaries' results are not as relevant as those found in paid dictionaries or on Naver, Goo, Eijiro, and other web dictionaries.
Japanese English Dictionary & Translator Box / 英語日本語辞書 (iOS)
This is a multilingual dictionary, geared toward Japanese and English (but with many more dictionaries to install), that searches multiple dictionaries for results. Generally has the most accurate results out of any free dictionary app. It includes web resources such as Naver, Wikipedia, and Google Images. You can select which dictionaries to search. Offline and online dictionaries are included. Free.
imiwa? is an English-Japanese/Japanese-English dictioanry using JMDict files based on Jim Breen's EDICT project. It includes furigana on all kanji, and can support handwritten input via the Traditional Chinese character recognition keyboard. This dictionary works without an internet connection. Free.
Eijiro on the WEB (iOS)
This app queries the Eijiro 英辞郎 dictionary on the web to search for English and Japanese words. It is particularly appropriate for phrases and colloquial expressions. This dictionary requires an internet connection. Free.
Tangorin Japanese Dictionary (Android)
The paid offline version of the online dictionary Tangorin, searchable in Japanese and English.
Android Free Offline Dictionaries
There are many Japanese-English dictionaries for Android. Here is a selection.
NAVER Dictionary (Android, iOS)
This is a primarily Korean dictionary that nevertheless includes 13 languages, and has Japanese dictionaries. It requires an internet connection and can be slow because of the lookup time over your data connection. Free.
MultiDict searches Naver, Goo, and Weblio for English and Japanese words. Free.
Webdict searches several online dictionaries, including Eijiro, Wikipedia, and LONGMAN. Requires an internet connection. Free.
The iPad and iPhone come with free built-in Japanese dictionaries, including J-J, J-E, and E-J. To install them, highlight a word and choose "define" from the pop-up menu. Then go to "Manage" and press the cloud icon next to the Japanese dictionaries. Wait for them to download and install. These dictionaries include a full version of Daijirin, although it does not have the jump or handwriting functions that the paid app does. With these dictionaries installed, you can define words in Japanese and English from the "define" function when you highlight a word in Japanese on your Apple device.
Kenkyusha Shin Wa-Ei (7-ban) Ei-Wa (5-ban) Chūjiten 研究社新英和（第７版）和英（第５版）中辞典 (iOS)
This dictionary includes the latest versions of Kenkyusha's Japanese-English and English-Japanese dictionaries, with accurate results and an abundance of natural example sentences. Its major drawback is that while the interface and searching capabilities are advanced, it has no handwriting recognition. Compared with other JE/EJ paid dictionaries on the market, this is the highest quality. Free JE/EJ dictionaries generally do not compare. Paid.
Kenkyusha Shin Wa-Ei (7-ban) Ei-Wa (5-ban) Chūjiten 研究社新英和（第７版）和英（第５版）中辞典 (App Store; iPhone and iPad)
Most paid Japanese dictionaries are affordable and have features not available with free dictionaries, such as images, jump function, and extremely good handwriting recognition. Save for several dictionary packages for Android, they are less than $50 each and much more affordable than the average electronic dictionary; thus, they are worth the investment.
Ultra Sōgō Jisho 2013 ウルトラ統合辞書2013 (Android)
This app contains a variety of dictionaries and encyclopedias, including full color images, such as 広辞苑, several Japanese-English and other European language dictionaries, 新漢語林MX, 角川類語新辞典, a contemporary usage dictionary, and more. It has the option to simultaneously search all dictionaries, or to choose several or one to search. Paid.
Kōjien Dai-roku-ban 広辞苑 第六版 (Android)
Kōjien is the gold standard of Japanese-Japanese dictionaries, and this version includes color images. Does not require an internet connection. Paid.
Daijisen 大辞泉 (Android, iOS)
Daijisen is a free dictionary with a paid option (paid only on Android), and is a comprehensive Japanese-Japanese dictionary including some images. It does not require an internet connection even for the free version, but the number of searches per day is limited without the paid version.
Daijirin 大辞林 (Android, iOS)
Daijirin is a comprehensive Japanese-Japanese dictionary, including images for some entries, and facilitates sharing words via Twitter. It includes a jump function, handwriting recognition, and browsing. Does not require an internet connection. Paid.
Iwanami Kokugo Jiten Dai-nana-ban Shinpan 岩波 国語辞典 第七版 新版 (Android, iOS)
Iwanami's Kokugo Jiten 7th Edition, similar in interface to Daijirin. It includes browsing capability and handwriting recognition. No internet connection required. Paid.
Sanseidō Kokugo Jiten 三省堂国語辞典 (Android, iOS)
Sanseidō's Kokugo Jiten is available as both an Android and iOS (iPhone and iPad) app. It does not include images, but allows for browsing and highlights search terms within words. Paid.
Meikyō Kokugo Jiten Dai-ni-han (Android, iOS)
Meikyō's Kokugo Jiten 2nd Edition, available for both Android and iOS. It supports handwriting recognition, and includes a kanji dictionary and several appedices containing information about topics such as lunar cycles. Paid.
Shin-Meikai Kokugo Jiten 新明解国語辞典 (Android, iOS)
This is a paid, cross-platform 国語辞典 that is the digital verison of the print 新明解国語辞典. It's available in both 6th and 7th editions from several distributors with different interfaces. Paid.
Ōbunsha Zenyaku Kogo Jiten (Dai-san-ban) 旺文社 全訳古語辞典 (第三版) (iOS)
This classical Japanese dictionary supports handwriting recognition and has a jump function. It also has full-color images and appendices such as the Hyakunin-isshu, and does not require an internet connection. Paid.
Weblio Kogo Jiten 古語辞典 (Android)
This is a free classical Japanese dictionary for Android powered by Weblio. Requires an internet connection.
Kanjigen 漢字源 (Android, iOS)
Kanjigen is a cross-platform kanji dictionary that includes search by keyword, part of kanji, or index. Paid.
Shin-Kangorin Dai-ni-han 新漢語林 第二版 (Android, iOS)
This is a Chinese-Japanese dictionary, focused on reading kanbun, that includes handwriting recognition, usage examples, and examples from the Analects and Shi Ji. It also provides the stroke count and order for various kanji. It includes 14,629 kanji and around 50,000 jukugo. Paid.