Books
Explained as well as I can.
Thanks to ccat for info and testing.
OD
Visit him and talk with him to start borrowing books.
Number of Books
The basic principle is that the first time you speak with him he will let you borrow 1 book. Once you've walked around enough different rooms and areas in the castle, and talk to him again, he will let you borrow another, for total of 2 books. And same again for walking around more different rooms and areas, for a total of 3.
How this works
Entering new areas/biomes enough times and go back to OD and talk with him again. It seems the maximum needed area/biome change is 2, and you can do this with just using the teleport.
Save before OD
Talk with OD (book1)
Use the Warproom, and warp to random 2 other warp rooms and back
Talk with OD (book2)
Use the Warproom, and warp to random 2 other warp rooms and back
Talk with OD (book3)
This means you can easy get all 3 books before fighting Andy.
Not a bug, NG+ will always set your books to 2, from the first time you talk with OD, and this can't be changed/increased.
Defeating the optional boss OD, will let you take all the books. (21)
Bugs:
If you trigger the “Talk about Train Pass” event with OD, when you have only 1 book, you'll get book 2, but won't get book 3 afterwards.
This can be worked around in some cases, the easiest is to go visit the train station before talking with OD first time. Then Talk with OD for book 1, and talk with him again straight away to get the “Talk about Train Pass” event out of the way. You can then get book 2+3 the way listed above.
(There is still a bug that could leave you at 1 book total, that we haven't figured out how to trigger yet)
The Books
Straight forward, as you uncover more map percentage, OD will have new books that you can borrow. This is completely independent of the number of books you can borrow at the same time (above). You could have 100% map completion, and still only be allowed to borrow one book at a time (Through either avoiding OD or a bug).
Might Tome
Sage's Tome
Sentinel Tome
Believer's Tome
Fortune Tome
Coldstave Tome
Heatstave Tome
Shockstave Tome
Empyreal Tome
Obscurity Tome
Corporeity Tome
Immunity Tome
Blessed Tome
Prowess Tome
Ancillary Tome
Deadeye Tome
Tailwind Tome (50%)
Tactician's Tome (60%)
Mastery Tome (60%)
Godsend Tome (70%)
Tome of Conquest (99%)