Moving this discussion to the Adobe Digital Editions forum.
Did you purchase the book from Barnes and Nobles? If so then you will want to download the book directly on the Nook. You will not be able to open the purchased book in Adobe Digital Editions. Instead if you were wishing to open the book on your computer you would use the Nook application. This application however does not provide sync capabilities for the Nook. Instead you would want to download the book on your Nook directly.
But HOW do I download directly to the Nook???
Locate a place with WiFi and download the book there while you have access. Most public libraries provide this service as well as many retail establishments such as coffee shops or restaurants.
I did find a discussion on Barnes and Nobles support site which details sideloading the book from your PC. You can find this discussion at http://bookclubs.barnesandnoble.com/t5/NOOK-First-Edition-General/how-to-transfer-book-fro m-PC-to-nook/td-p/750902.
If you have additional inquires on how to get your eBooks on you Nook I would recommend working with Barnes and Noble directly.