As AnemicOak says, the Kindle is not a good choice for library books; you want an device that supports ePub books, since most library books will be in ePub format.
The way it works is that you "check out" a book and download it to your reader. You can read it for typically two weeks, after which time it "expires" and stops working. If you finish it more quickly, you can "return" it to the library, thus allowing you to check out another book more quickly.
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