B&N ebooks from a few months ago would have been in the eReader format (file extension .pdb). If you use B&N's desktop reader (Windows or Mac) to re-download them today they might be in ePub format instead (file extension .epub). Under Windows the new ebooks end up in a sub-folder of "My Documents\My BN eBooks" named after your e-mail address.
Either kind of ebook will have DRM (encryption) based on your credit card name and number. Both can be removed, but under Windows at least it is easier to strip the DRM from B&N ePubs (see
Barnes & Noble DRM for EPUB circumvented). For eReader DRM, you would use ereader2html.py from the command line. For both ePub and eReader you will need to install python, see
Visual Kindle Guide for how to do this and also how to use the "Command Prompt" which is only needed for eReader (the ePub script works by clicking on it, like standard Windows apps).
Once DRM-free, use Calibre to convert to MOBI (with the Kindle target). Note that removing DRM and format shifting for personal use (e.g. to use a different kind of reading device) is probably legal almost everywhere, including in the US.