What is the exact error message you are receiving when you try to sync the books to your Nook? Also which operating system are you using?
Unauthorise the computer using ctrl-shift-D to ADE.
You should then be able to reauthorize it correctly.
If you have incorrectly authorized any devices, deauthorize them using ctrl-shift-E to ADE when they are attached..
I reauthorized my computer but my Nook HD doesn't show up in the library to transfer ebooks.
From thread http://forums.adobe.com/thread/1128760?tstart=0 : post 2
If you goto the settings on theNook HD or HD+ and select All Settings and then Account Settings there is a place to log right into ade. BN made it so simple! Hope this helps! LD
Also, in thread http://forums.adobe.com/message/4968300#4968300 another user posted
I corrected my problem by having staff at Barnes & Noble upload the Digital Editions app to my Nook.
That makes is sound as if there is a Digital Editions app available, but not always pre-installed.
There is also a much longer and possibly incorrect post from me on that thread. http://forums.adobe.com/thread/1128760?tstart=0 post 1
That post is probably wrong becuase it sounds as if the Nook tablets come with some software that means they can behave like a dedicated eReader;
If any of those helps for you, please let us know how you resolved it so we can give mopre precise adcive in future.