In Iceland, most often a person is given first name(s) ("eiginnafn") one or two or in rare cases more, followed by surname ("föðurnafn" or "móðurnafn") which is most often made up of the father's name ("föðurnafn") or in rare cases the mother's name ("móðurnafn").
For example my "eiginnafn" are Örn Albert (two first names) and my "föðurnafn" is Þórarinsson because my father first name was Þórarinn. The "föðurnafn" is therefore made up of Þórarins (that is Þórarinn in genitive case (one of four cases for icelandic nouns)) followed by "son" meaning "son of Þórarinn" or Þórarinsson" which can also be written in english as "Þórarin's son".
My son name is Þórarinn Örn Arnarson with his "eiginnafn" being "Þórarinn Örn" and his "föðurnaf" being Arnarson (Arnar is the genitive case for Örn followed by son) meaning son of Örn.
Some icelandic people have family name, often added after or before the "föðurnafn" for example Jón Metúsalemsson Kérúlf - where Jón is the "eiginnafn" Metúsalemsson" is the "föðurnafn" and Kérúlf is the family name.
If I had a daughter with the name Arna Dís, then her "föðurnafn" would be Arnardóttir, which means daughter of Örn fromed with Arnar (genitive case for Örn) followed by dóttir which means daughter. This could be written Örn's daughter in english.