How do I read a borrowed book on my Nook?

You must activate the Nook.  What sort of a Nook?

If a standalone eReader that shows up in the left pane of Digital Editions, then use menu/Help/Authorize Device (Library/Authorize Device on the older more reliable v1.7.2) to ADE with the device plugged in.

Otherwise, read the documentation for the Nook or for the eReader app you are running on it.

[moved]

There is a special subforum for the Android version of Reader

http://forums.adobe.com/community/adobe_reader_forums/android?view=discussions

although I think that your question is more related with Digital Editions,

http://forums.adobe.com/community/adobe_digital_editions