I believe in your collections, Books refers to the purchases books which are on your device, Purchased books are the ones that you paid for or got because they are free and may or may not be on your device, and PDF are external files that you transferred into iBooks for reading.
One problem I have found is that if you get a pdf from another app and transfer it to iBooks, it will not be saved in a backup. So when you restore those PDFs will be missing. You can always get your books that came from the iBookstore back, but unless you saved those PDFs on your computer some place, they will be gone. It is likely best to save your PDFs on your PC and then transfer them to iBooks via iTunes, to ensure they are backed up.
Another problem I have with iBooks, those I like it has an app, is that AFAIK, you can only read your books within this app! That means on an iDevice (I have not checked if iBooks in available on the Mac OS....that, IMO, sucks. If you get a Kindle book, you can read it in some many different places that the mind boogles. So I tend to get those instead.
If I am wrong in any of this, someone please correct me.