I've just spent a couple of days working in eBook readers for my Android phone. I've tried Moon Reader, Aldiko, FBReader, Wordaholic, WordPlayer, and perhaps one or two more I've already forgotten.
None of them, possibly with the exception of FBReader, seems to automatically import eBooks that are just placed directly onto the Android phone. They want you to fiddle around to import the books; some readers come from companies that quite explicity expect you to purchase books from their online stores.
I'm going to give FBReader another chance since it did manage to read my directory file automatically this time. Maybe I'll have better luck with it. Other readers were more polished, but I don't need the headache.
As for what all this means, the lowest common denominator in this foodfight is the format of the eBook itself, and the winning format is "epub." The web site that offer books for free download seem to be standardized. Other than that, I'm not certain what's happening other than a mad scramble for revenue.
I will say the the free, open-source, user-supported software "Calibre" to manage your eBook library works just as advertised. And when you push it to its limits, like I did, the developer will step in to support you. Give it a whirl.