Must I pay for a book that someone claims I borrowed and did not return, even though I do not remember doing so and the book is not in my possession? And what is the ruling on compensation in general?

This person who made the claim is responsible for providing proof. If he has no proof, he may ask you to swear an oath. If you swear the oath, he has nothing against you except your oath, and you are not obligated to compensate him for anything. However, if you believe that this situation could cause animosity between you, and you are able to compensate him, then do so out of kindness and to preserve your affection for each other.