If you intend to stay in Iceland longer than three months and you are a citizen of a country that is not an EEA/EFTA member state, you need a residence permit. You can read more about that here.

A residence permit based on family reunification is granted to the closest relative of a person residing in Iceland. The requirements and the rights of residence permits on grounds of family reunification can differ, depending on the kind of residence permit applied for.

Information about residence permits based on family reunification can be found here on the website of the Directorate of Immigrants.