Aozora Bunko 青空文庫 is a free digital library of modern, public domain literary and non-fiction works in Japanese (including some translations into Japanese). They are available as plain text or XHTML with ruby (glosses). Aozora texts can be searched, downloaded, and read through several apps for both iOS and Android.
青空キンドル is a website that converts Aozora files into appropriately-sized PDFs for the Amazon Kindle. You can choose text size of small, medium, or large (large is recommended). To use it, simply upload the .zip file for the work that can be downloaded from Aozora's site.
neo文庫 is a paid app for iPad and iPhone that syncs your reading progress in texts between iPad/iPhone via iCloud. It includes a ranking of Aozora's top 500, search by author and work title, and genre.
Hōhei Bunko 豊平文庫 (iOS)
豊平文庫 is an iPhone app for reading Aozora books, available in both free and paid versions, and is intended for users using iOS4.3 or newer. (neo文庫 is recommended for iOS5 and above, and for iPad.) It integrates with Daijirin and Daijisen dictionaries and provides search by author, work, genre, and top 500.
iBunkoHD is an e-reader for iPad that supports both vertical and horizontal text. Although it comes pre-loaded with 230 books from Aozora Bunko, it can support any text, PDF, CBZ, or CBR file. Paid.
Yomizukue よみづくえ (iOS)
Yomizukue is an Aozora reader for iPad. It comes loaded with 100 books from Aozorar and includes Daijirin and Daijisen dictionary integration. Paid.
Aozora Yomite (Android)
This app connects directly to Aozora Bunko and allows access to rankings. Free.
Aoi Sora (Android)
This app comes in both free (ad-supported) and paid versions. It is suitable for Android tablets as well as phones. Free.
Vertical Text Viewer (Android)
This app is a simple viewer for Aozora Bunko books. It cannot directly download from Aozora. Free.