Names are remarkably complicated things. From my earliest years I was aware, courtesy of my dad, that our family name, Schuler, was German in language and Swiss in national origin. My first name, David, is a family name, too. It’s my great-great-grandfathers name. Among the Swiss it used to be a commonplace for all of the sons of a family to be given the same first name and different middle names, e.g. my grandfather and his brothers were Joseph Melchior, Joseph Leonard, Joseph Louis, Joseph Anton, Joseph Jacob, and Karl. I don’t know what happened with Karl. He died as an infant so that have had something to do with his being shortchanged in the name department.
My dad also told us that German family names were adopted beginning in the Middle Ages, about the same time as they were adopted in England and France. Our Swiss Schuler family name, for example, was in use by our family as early as the 14th century. Family names were made mandatory in the Austrian Empire by Emperor Josef II in 1787 when he commanded that all Jews within his realm adopt German surnames and that place names were forbidden to them.
Most family names in Europe are patronymic in origin, that is they are derived ultimately from somebody’s father’s name. Johnson, Jansen, Johansen, Jones, MacIan (also McCain), Giovanni, Ioannidis all mean son of John and in most European countries half or more of all family names are patronymics. Iceland retains true patronymic family names to this day, i.e. if your father’s name was John your surname is Johannsen if you’re a man and Johansdottir if you’re a woman. I understand that the Reykjavik phone book is arranged by given name.
Other sources for family names are occupational names (Smith, Schmidt, Favre, smith or Taylor, Schneider, Thayer, tailor), place names, and even characteristics. Reid and Roth are both names for redheads.
In Europe most family names are surnames, i.e. they are placed at the end of the name although even in Europe that’s not universal: in Hungary family names are placed at the beginning of the name. In China family names are placed at the beginning, too.
Russians have three names: a given name, a patronymic, and a family name. Ivan Ivanovich Ivanov is John son of John of the Johnses. His friends call him Ivan Ivanovich. I began studying Russian when I was 13. It was my first introduction to culturally different naming conventions.
In Spain and many Spanish-speaking countries in addition to their given names people receive two surnames: one from their father’s side and one from their mother’s side. The given name occurs first followed by the father’s family name then the mother’s family name although mother/father is also used.
Arabic names consist of a given name, one or more patrynomics, and an occupational, geographic, or tribal name. Additionally, if a man has a son, his son’s name may occur before the other names. There may also be nicknames or honorifics. So, for example, Abu Kareem Muhammad al-Jameel ibn Nidaal ibn Abd al-Aziz al-Filisteeni indicates that its bearer is the father of Kareem, has a given name of Muhammad, is called the handsome, his father’s name was Nidaal, his grandfather’s name was Abd al-Aziz and he’s from Palestine.
I know nothing whatever of African naming conventions. My understanding is that, although family names are common they’re not universal and although they are most commonly surnames, that’s not universal, either.
Although all sorts of things can be inferred from naming conventions, some of them pretty far-fetched ideas of the deep cultural significance they bear, I think that mostly what naming conventions tell you about is history. Cultures with historic connections to one another tend to have related naming conventions, too.
Update
I just realized that I inadvertently omitted a major source of family names. In addition to patronymics, occupational names, and geographical or topographical names there are also patronage names. In peasant and slave cultures it was a commonplace for the serfs or slaves of a patron to take the patron’s family name as their own. It is believed that the surnames of most African Americans are patronage names.