I'm going to share from the perspective of Auckland City Library - as this is the library I use. However, a quick google, led me to an article saying around 40 NZ libraries had signed up to using Overdrive as one of their core apps (there are several others) to deliver eBooks through. All you need is to be a member of the particular library in order to issue books.
To use Overdrive, you simply need to download the app which is available on apple and android, as well as a web version. From here you need to add a library through the side bar, log in using your library card number and you are away. The app is simple to navigate and move through. The tool bar on the left hand side gives you access to your library, bookshelf and more.
When searching for books a small symbol on the top right hand corner tells you automatically if they are in (black book) or currently issued (grey book).
By tapping on the book cover, it takes you to more in-depth information about the book where you can download a small sample of the book, read a blurb, see how many copies are available and how many 'holds' there may currently be on a title.
If the book is currently issued you are able to reserve a book by using the 'Borrow' function. You are able to change options easily and see how many people are ahead of you in the queue before you can get your hands on the book. The brilliance of this 'holds' area is that you can set up an email alert to let you know when the book has become available for you. You can set the book to automatically be added to your shelf when it is yours to read. Very handy!
The account area gives you a quick glance to where you are for issues, holds and how many books are currently available from your wish list too.
I know I have enjoyed watching the range grow and develop. Overdrive is also involved in the library Big Read - a global event which happens once a year, where 2 books for children/teens are available to read. People are also invited to take part in the discussion through twitter and blogs.
Just a quick overview of the Overdrive app - my students have enjoyed beginning to use it and the luxury of having free eBooks at their fingertips. Happy reading!