To get epub or mobi files opening in the Amazon Kindle app on your Android tablet you will need to do the following:
Install Calibre on your PC (most O/S supported)
http://calibre-ebook.com/download
When l do this on windows the first time you open Calibre it asks you what type of Reading App you are using and on what type of device. I selected Amazon Kindle Reader on an Android Tablet.
You can then drag and drop your files into Calibre.
Connect your android tablet to the your PC. Calibre should recognise it and when you right hand click on a book your should have the option to transfer the book to your tablet.
For some reason the book will not instantly appear in the Kindle app, you will need to give it a few minutes, sometimes even force the app to close
Happy reading