Nordic citizens living in Sweden
If you are a citizen of Denmark, Finland, Norway or Iceland you may be considered both a Nordic citizen and an EU citizen, and use two different sets of rules.
Denmark, Finland, Norway and Iceland, along with Sweden, constitute the Nordic countries. The Nordic countries have an agreement which states that Nordic citizens have a right to live in whichever Nordic country they choose. You do not need to apply for a residence permit. If you want to live in Sweden you simply need to register with the Swedish Tax Agency.
If your family are citizens of a non-EU country and you intend to live in Sweden, they can apply for residence permits in accordance with Swedish rules.
EU citizens and their families have a right to move to another EU country as long as certain requirements are met. An EU citizen has the right to live in an EU country which is not their country of origin as long as they meet the requirements for right of residence through work, studies or with sufficient means. In Sweden you simply need to register with the Swedish Tax Agency.
As a Nordic citizen you are also considered as an EU citizen. You do not need to apply for permits to live in Sweden. If your family are citizens of a non-EU country they can live with you in Sweden as long as you have right of residence. In this case, your family should apply for a residence card in accordance with EU rules.