This fatherly adjective means “anything based on relationship to the father or descent through the male line.” While it seems like a distinguished, old-fashioned term, it’s a fairly new word. It was coined in the 20th century, and the usage has been almost exclusively for anthropological and sociological studies. Someone wouldn’t talk about their dad and refer to their “patrilineal relative,” for example. “Patri-” comes from the Latin word for “father,” “pater,” and the corresponding adjective from the Latin for “mother” (“mater”) is “matrilineal.”