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.

Student’s residence permits are granted to the following:

• Individuals intending to engage in full time studies at a University in Iceland.
• Post graduate students from foreign Universities collaborating with an Icelandic University.
• Exchange students from acknowledged exchange-student organizations. 
• Interns.
• Students in technical studies and recognized workplace studies at higher-education level.
• Graduate looking for employment.

Requirements, further information on documents to be submitted and an application form can be found on the website of the Directorate of Immigrants.