The damaged book is a book held in the pocket slot that can be obtained for 5,000 coins from Jossik after completing Horror From The Deep. The four Guthix pages can be added to upgrade it to Guthix's Book of Balance, which gives a damage bonus. Dropping or destroying a damaged book and talking to Jossik results in him returning the book.
Players can check which pages are currently in your incomplete book with the "Check" option. Filling the book with all pages currently costs 422,616 coins. Once a page has been placed in a book, it cannot be removed.