From the Amazon website:
Borrow Kindle Books from Your Local Library
You can check out a Kindle book from your local library and read it on any generation Kindle device or free Kindle reading app.
When you borrow a Kindle public library book, you’ll have access to all the unique features of Kindle books, including real page numbers and Whispersync technology that synchronizes your notes, highlights, and last page read. After a public library book expires, if you check it out again or choose to purchase it from the Kindle store, all of your annotations and bookmarks will be preserved.
Kindle books are available at more than 11,000 libraries in the U.S.
How It Works
You can borrow Kindle books from your local library’s website and, with the click of a button, have them delivered to your Kindle device or free reading app.
• Visit the website of a U.S. library that offers digital services from OverDrive.
• Check out a Kindle book (using a valid library card).
• Click on “Get for Kindle” and then sign in to your Amazon.com account to have the book delivered to your Kindle device or reading app.
Note: Public library books can be sent wirelessly to Kindle devices via an active Wi-Fi connection or transferred via USB.