For the library OverDrive books, you'll need to download Adobe Digital Editions, set up an Adobe username and password, authorize your Adobe Digital Editions software and then with the software open, plug your Nook into your computer. The Adobe Digital Editions software should recognize it, ask if you want to authorize it, and once you say yes, the Nook should let you put downloaded Adobe PDF and Adobe EPUB books onto it. You should then just be able to download them from the library website, open them in Adobe Digital Editions and send them to the Nook. Different libraries have different variety of e-books to offer. I know mine offers a very random selection. Adobe Digital Editions is good to have anyway, because it lets you buy Adobe DRM e-books from other stores like Fictionwise or Kobo or many others.
As for other free e-books, it depends on what type of book you enjoy. I have found the vast majority of free e-books are romance novels or public domain.
For free public domain books, I first like to search
Feedbooks, and then if they don't have it, I go to
Project Gutenberg. These tend to be much more readable than the free classics you can download from B&N or Google Books because they weren't just scanned in and converted to e-books, they were proofed by real people.
A nice search engine that can be used to find free e-books is
Inkmesh. There's a link for "Free Nook Books," which I like to check every few days because it helps you see if anything new has been listed. This search engine also lets you search for non-free books and price compare. Sometimes buying direct from B&N is not the best way to go. If you're willing to go through the hassle of downloading the Sony store software, the free Sony books also work on the Nook (and you also need the Adobe Digital Editions software that's needed for the Overdrive library books).
If you like scifi and fantasy, Baen offers a bunch of free e-books at their
Free Library.
If you're willing to sift through the offerings, you can also potentially find some good free and low-cost e-books at
Smashwords. This is
mostly independent authors, so there's a fair amount of..."less good" stuff , but it's also possible you might find a gem, and some of the e-books there are by traditionally published authors who got the electronic rights to their books back or never sold them in the first place who make their own e-books of their traditionally published books.
If you like romance,
Mills and Boon and
Harlequin offer a handful of free e-books too. There is overlap here between what's on these sites and what's free from B&N, but there are some different ones too.