ADE works differently with tablets (such as Nook HD) and dedicated eReaders (such as maybe your previous Nook?).
Supported dedicated eReaders show up in ADE on PC/Mac as a device, which you authorize for Adobe DRM from ADE, and then copy files using ADE (or other route).
That's the part you are already used to.
Tablets need an app running on the tablet: I suggest you try Bluefire (but see caveat below).
Bluefire will let you do the AdobeID authorization directly from the Bluefire app on the tablet.
You then copy files from PC to tablet using Windows File Manager or whatever, and read them using Bluefire.
There is a dedicated app for Overdrive library books that allows you to borrow and read books without going via the PC al all.
Similarly, several publishers have apps that let you buy and download books directly using the tablet.
For non-DRM books, no need to worry about Adobe software or authorization at all;
just copy them to the tablet (or download them directly from the internet) and read them with Bluefire or whatever.
Some tablets are sold subsidized and then have restrictions on what apps you can install
(trying to force you to buy books from the tablet supplier).
I'm not sure exactly what restrictions (if any) there are on the Nook HD.
If all the above makes sense and you get things working,
please post back what did and didn't work and what restrictions you found;
that will help other forum readers and allow us to give better advice in future.
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