I am a citizen of a country outside EEA/EFTA
A residence permit is required for citizens outside the EEA/EFTA area and EFTA who intend to reside in Iceland for more than three months. The Icelandic Directorate of Immigration decides whether residence permits are issued.
All persons who are issued a residence permit for a stay longer than 3 months should have a legal domicile in Iceland. When issuing a residence permit, the Icelandic Directorate of Immigration sends Registers Iceland a request for a legal domicile to be registered in the National Registry, along with the additional documentation required for registration in the National Registry.
All persons who register a legal domicile in Iceland for the first time are listed in the National Registry and allocated a personal identification number. Persons who have once been issued a personal identification number keep it for their rest of their lives even if they move from Iceland. An ID No. without a registered legal domicile in Iceland provides limited rights.
For further details, see:
Notification of residence in Iceland (foreign citizens) - The form shall be handed in to the Directorate of Immigration
What is Registers Iceland, and what information about me is registered?
Posting.is - Information on the rights and obligations of foreign service companies and their employees who are posted temporarily in Iceland