Is this a Nook eReader with books downloaded via Adobe Digital Editions (ADE), or a standalone tablet.
For an eReader, while it is plugged in to the computer, run ADE and choose menu/Library/Authorize Device (ADEv1.7.2) or menu/Help/Authorize Device (ADEv2.0).
For a tablet, make sure you have authorized the tablet, using the ebook app on the tablet.
Some people have problems authorizing Overdrive app on a tablet.
In that case, try installing Bluefire app and authorizing that.
It may even be that authorizing Bluefire permits Overdrive to work, as it is the device rather than the app that actaully gets authorized.
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There are lots of bugs in ADE2.0.
One of these is that it does not work with Overdrive library books on some devices.
The problem devices include many Sony devices, but I understand there are others problem devices as well.
Try replacing ADE2.0 with the older but more reliable v1.7.2.
(You can have them both installed at once if you like.)
Version 1.7.2, it is a little difficult to find, available on Adobe site for Windows and for Mac.
http://helpx.adobe.com/digital-editions/kb/cant-install-digital-editio ns.html
The forum software is sometimes corrupting the link above. There shouldn't be a blank in 'editio ns.html'. The following redirects to the same page: http://tinyurl.com/diged172