Non-EEA citizens need a residence permit (dvalarleyfi) issued by the Directorate of Immigration (Útlendingastofnun) on a specific basis, work, study, family, or international protection. The permit must be renewed on time. After roughly four years of continuous, legal residence on the same basis, you can apply for permanent residence (búsetuleyfi). EEA/EFTA citizens do not need a permit; they register their right of residence and gain a permanent right after five years. If a decision goes against you, you can appeal to the Immigration Appeals Board (Kærunefnd útlendingamála) within a short window.
A note on prices: residence permits, cards and appeals all carry processing fees, and those fees change. Wherever a fee is involved, this page links straight to the official page so you always see the current amount, never an outdated one from us.