Icelandic is an Indo-European language, belonging to the group of North Germanic languages, to be specific. This group also includes Danish, Norwegian, Swedish, and Faroese. Of those languages, Norwegian and Faroese (spoken in the Faroe Islands) are the most closely related to Icelandic.
Icelanders and Faroese people may be able to understand each other’s languages on the page, as their writing systems and spelling are quite similar. Speaking is another matter, however: the pronunciation differs significantly, and the two languages are not mutually intelligible without study.
Icelanders receive some help from their schooling when it comes to understanding the other North Germanic languages. Danish is a compulsory subject in schools, and learning it helps with the comprehension of Norwegian and Swedish as well.