For this thing, I first downloaded the Free Books app, which appears to be called simply 'Books' nowadays. Due to the name change I had a heck of a time, first finding it in the app store, and then being certain that I had the correct app. I poked around a bit, downloaded a sample ebook (Benjamin Franklin's autobiography) and paged through it. Overall, the interface seems pretty slick. While I'm not a big consumer of ebooks in general, I could easily see myself using this to read free classics while on vacation or in another situation in which I'd rather not be lugging around a print book.
Because I have a 4-year-old child, I was very interested in trying out the iStoryBooks app, but to my disappointment it appears to have been pulled from iTunes. (I'm uncertain whether it is still available to Android devices.) I attempted to find it via both a general search and a direct link from the 23 Mobile Things website. Both came up empty. If I ever find it, I'll post an update.
I passed on the Wattpad, Audiobooks and YALSA Teen Book Finder apps since I'm less interested in these topics.
My go-to app for interacting with other readers and gathering ideas for new things to read is GoodReads. I've never checked to see whether it offers downloadable, copyright-free titles, so I can't comment on that aspect, but I love that I can see what my friends and family are reading, keep track of what I read, and share ratings and reviews.