Should I mention a mistake made by a third person? 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.