The Icelandic telephone book (símaskrá)for instance, is organized after first names. If your first name is Aase,you will be first in the phone book.
Some Icelanders do have family names but they are listed by their firstnames in the phone book.

Instead of family names, Icelanders use the first name of their fathers,and add -son if they are male and -dóttir if they are female.
If your name is Anna, and your fathers name is Jón, you willbe called Anna jónsdóttir (Anna Johns daughter).
If your name is Gunnar and your fathers name is Þór, youwill be called Gunnar Þórsson (Gunnar Thors son)

This is the ancient viking name order, even used among the ancient Anglosaxons.

F.A.Q.
1) How do you tell different people with the same name apart?
Answer: 
It is no more difficult than telling all the John Smiths in New Yorkapart!
We often add peoples occupation or their address. Jón Jónsonthe plumber at Nesgatan is easily identifiable.
2) What happens with children, if you don´t know who the fatheris?
Answer: 
We find a fictive fathers name for them. Some people have been calledHansson or Hansdóttir (His son or His daughter), Karlsson or Karlsdóttir(Mans´ son or Mans´daughter) 
In ancient times some people were called their mothers sons.