Okay kiddo, a nomen gentilicium is a fancy way of saying a person's last name in ancient Rome. Back then, people didn't have just one name like we do now, they had three! Let's use a guy named Marcus Aurelius as an example.
His first name was Marcus (like mine!), his middle name was his dad's first name, and his last name was his family name. So his full name would be something like Marcus Julius Aurelius. The "nomen gentilicium" part is just the last name - "Aurelius" in this case.
It was really important in ancient Rome to know someone's full name because that's how they kept track of families and social status. So if you met Marcus Aurelius on the street, you'd want to know his full name so you could figure out which Aurelius family he belonged to.
Nowadays, we usually just have one last name and we don't use our dad's name as a middle name all the time. But the idea of a nomen gentilicium is still around in some cultures!