I downloaded the Kindle app to my Android phone the other day and on the whole it works fine. However, it is supposed to open your book where you left off, even if you left off on a different device. I haven't been able to get this to work yet. I was reading a book on my Kindle and when I opened it on my phone, it asked me if I wanted to sync. When I did it was about 2 chapters behind where I had left off. And I have yet to have it sync on the Kindle. I'll keep trying.
I also downloaded the new Borders app on my phone but haven't had a chance to play with it yet.