If your last name was handed down from your father or his ancestors, you can call it a patronym. Across the world, patronyms are the most common type of surname.
Inheriting your father's last name doesn't make it a patronym — instead, patronyms add a suffix or prefix that means "son of." Common surnames like Johnson and Fitzgerald are patronyms, having the literal meanings "son of John" and "son of Gerald," respectively. Patronyms are still used in Iceland, where if your dad's first name were Grímur, your last name would be Grímsson. Patronym combines Greek roots pater, "father," and onym, "name."