Once your library e-books are in the Adobe Digital Editions program, you can easily transfer them to your e-reader:
Be sure you are in "Library View" by clicking on the icon with 3 books in the upper left:
Connect your e-reader to your computer so that you see it listed in the "Bookshelves" area"
Drag and drop the e-book's cover to the name of your e-reader device; the transfer will take a few seconds. It should then be ready to read on your e-reader. (To drag and drop, click and hold your left mouse button on the e-book's cover, drag your mouse to the name of your e-reader and let your mouse button up.)
Where are they? Files that you download yourself (not through the e-reader's bookstore) may not display in the same location on your e-reader. If you cannot find them, try these:
For Nooks, look in the My Documents folder, not the My B&N Library folder