In the West, the family name is treated as a surname and called a last name. Family names under the eastern order come before proper names. The eastern order refers to naming in East Asia, where emphasis is placed on the family and not the individual. (Note: Hungary also follows the eastern order.) I would not want my last name to come first, because it doesn’t sound great in western order. Thus, my western cultural sensibilities lead me to believe that it would look and sound awful in the eastern order also. Despite that fact, I do admire the honor that it bestows upon a person’s ancestry. In addition to serving as a link to ancestry, in English speaking countries it can also serve as an indicator of occupations (Smith), personal characteristics (Black), geographical features (Hill), locations (London), estates (Windsor), and patronage (Kilpatrick- follower of Patrick). If you want more information on this topic, it is all covered in the family name Wikipedia article.
However, there is a large outlier in the West and in the context of family names, Iceland. In Iceland, there are no family names, instead surnames are uniquely patronymic or matronymic. Icelandic surnames are derived generally from the immediate father, albeit there are a few exceptions where names are derived from the immediate mother. For example, when Karl is the father, Karlsdóttir is the surname of Karl’s daughter and Karlsson is the surname of Karl’s son.
Fascinating, but now there are several questions that arise: (1) Is it harder to trace a person’s genealogy? (2) How would you identify siblings? (3) Would you choose to share the same proper name with your son (Karl Karlsson)? (4) How do you find someone? Nonetheless, there is a little relief, the phone book goes by proper names first and then by profession. Still, that does not seem to solve the problem of finding a mechanic named Jón, who is the son of Daniel.