A professor asked from me to send one of his books I've had. Because I've had to leave for some days from the country I've arranged with a relative to send the book. However, he ignore it and now the professor sent me an email asking for what happened. In my reply should I mention that the fault is not mine?
It was your task to ensure the book gets returned. You need to take responsibility and apologize. You can mention that you had made arrangements, which failed unfortunately.
It's unlikely that the professor cares about the details. What he cares about is that he didn't get the book (that's why you apologize) and that he can rely on and trust you (that's why you promise it doesn't happen again and make sure it doesn't). This is a learning experience, i.e., you know now that your relative is not reliable and as a consequence your professor trusts you slightly less (but that's not permanent damage). Obviously, make sure he gets the book as soon as possible.
I disagree with the accepted answer. Yes, it was your task to return the book so, yes, you need to take responsibility and apologize.
However, you should also mention that you delegated the task to somebody else and they have let you down. Suppose you don't mention this, your relative doesn't send the book for another week and the professor sends you another mail. How are you going to explain that you still haven't done what you said you'd do?
And suppose that you later want to borrow another book. If the professor thinks it's completely your fault that the first book was late, they'll be reluctant to lend you another. But, "I'll never get that person to return another book for me!" is a concrete and believable plan for getting the next book back on time.
I believe too many assumptions have to be made in order for the above answers to be completely valid.
Starting from the moment the professor requests this book, he has sent you a task. Since you said you are out of the country, you can either be on vacation or on university work. If it's the first case, your responsibility to even answer in the first place is reduced unlike the second. Anyway, after you accept to do this, the task of getting the book to the professor is now your responsibility.
Since you are not physically able to perform the task, you delegated said task to another agent, which failed to complete it. Did you get initial feedback from the other person that they would follow up on the task? If so, they have drawn the responsiblity to themselves and then failed. If you never had any confirmation this would be performed, then you never passed on the responsibility to complete said task.
Since the professor asked you "what happened", if you think the guilt distribution on the third person is of relevant value, you shouldn't conceal it because the professor would be blaming you whilst you aren't responsible. If he "doesn't care" about the facts that led to the delay, he isn't a very reasonable person in the first place and he should apologise.