The Overdrive Media Console app is for one of the big library ebook sharing solutions. The Bluefire Reader app is for the other. Between the two they cover just about every library that has ebook sharing available.
As others have said, you'll have to check with your local library to find out which to use and how to get books onto them. If like ours, they've got a pamphlet somewhere.
Some libraries are also using the Kindle for ebook lending. Ours has a couple, but they are limited to sharing devices with preloaded books, not sharing through your own Kindle or Kindle app. That might just be here though. I'm not what the full extend of Amazon's libary involvement is.