In Spain they have two family names (same as Mexico): the father's first family name + mother's first family name. I think this becomes a bit flexible these days. In Iceland they take the father's first name and add "son" or "dóttir" (daughter). e.g. Jóhanna Sigurðardóttir would be a girl named Jóhanna, with a father named Sigurð. Her bother Helga would be Helga Sigurðson. In France, Germany, UK, etc we are approximately as Nanren888 describes. Traditionally, children take the father's family name, but this now becomes flexible e.g. maybe they take the mother's family name (if the parents are not married, or if the mother got married without changing her name). In some countries it is possible to have a double-barrelled surname, and so the whole family can combine both the mother's and father's family names e.g. "John Smith-Jones" (mother is Smith, father is Jones, for example). This is different to Spain/Mexico, because it is still one family name, but double-barrelled. I believe that this is not allowed for children in Switzerland. Then, in some countries you are allowed to have a middle name (or many middle names). The middle name can be anything, but sometimes people put names of relatives (first names or family names) to keep them alive. This is really common in the USA. Also special in USA is that people really use these names e.g. "John F. Kennedy", "George W. Bush". In the UK, we don't usually tell anyone about our middle names. In short, it varies in every country.

Adding something to Bill Holt's asnwer. Typically in the West; Children get the father's last name (surname, name of the family) When women marry, they typically change their last names (surname) to that of their husband. A minority don't. Children get the father's last name (surname, name of the family) '... repeat . Some choose to give their children a hyphenated name Holt-Grove, for example, if a Holt married a Grove. . With increasing incidence of people not formally marrying, we are becomming slightly more similr to the Chinese model, of the female partner not changing her name. This also happens often if the woman has a professional name for herself and does not want to lose reputation or confuse those who know her, such as a doctor or lawyer. . Generation names are essentially unheard of in the West. I suspect their use has dioed out in China too, as the names have mostly gone to two characters. . Three names, total are common, sometimes more. A first name, a middle name and a surname. The surname is shared by, the same for, all the family. It is still relatively common for first names, used in families to be re-used, or to name children after older relatives, such as a mother, father, grandmother or grandfather. My paternal grandfather's name was Edward Llewellyn <+surname>. He chose to name his son, Llewellyn Edward <+surname>. Similarly, just liking to continue a name is common. A girl might be named something new as a first name, but get given a middle name the same as that of a mother or grandmother. OF course the movies would have us believe that every thrid American has a name such as John Smilt Jr. (junior), [son named after father] or John Smith III, John Smith, the third, being the third in this family called exactly the same name.

Names are complicated. My last name is Holt. I inherited the name from my father; it has been passed from father to son for many generations since my ancestor Nicholas Holt immigrated to North America from England about 385 years ago. The name is an old English word that means "a small wood or grove of trees".