According to their promotional website:
The Bookshelf App itself is absolutely free. When you download the app, you’ll receive the scriptures, LDS study materials, and eight popular eBooks published by Deseret Book free of charge. At that point, you’ll have the opportunity to purchase additional titles through Deseret Book.What about my Kindle?:
Kindle owners: Don’t despair! If you prefer Kindle, you will notice that we offer over 1200 of our books for the Kindle format via the Kindle store on Amazon.com.It appears there are plans to release Deseret Bookshelf for Android, and if the rumors are true, you may be able to have it on the anticipated Kindle-tablet which would run off an Android platform.
However, I really wish they were able to deliver a Kindle e-ink/LCD hybrid as referenced (and shot down) in this article.
Finally, if you want the latest news on Deseret Bookshelf, check out their Facebook page or their site at DeseretBook.com
Update:
For those who own the Kindle Fire, be sure to check out instructions for installing
Update 2:
The same process applies to the free Deseret Bookshelf app; however, you'll need to locate the .apk file.
Update 5:
Desert Book has published an .apk here: http://deseretbook.com/misc/