You can transfer a checked out EBSCO eBook to your eReader (iPad, iPhone, Nook, etc.). EBSCO eBooks works with a variety of eReaders, except, unfortunately, the Amazon Kindle.
Here are the steps to download a book to your eReader.
There are a lot of great videos with tips on how to download eBooks to various eReaders. The following two videos, courtesy of University of Colorado Boulder libraries, show how to download to an iPad and a Nook. If you have another eReader, try searching YouTube for the "name of your eReader" and "download" and "ebooks."
Remember, at this point, EBSCO eBooks and Adobe Digital Editions do not work with the Amazon Kindle.