Scandinavian, or North Germanic, languages are from the highly influential and widespread Indo-European language family. Danish, Faroese, Icelandic, Norwegian, and Swedish are all considered to be Scandinavian languages.
By 1000 AD, the Danish, Norwegian, and Swedish languages had developed from dialects of Common Scandinavian. Today, speakers of those three languages can communicate with each other because of similarities in vocabulary and grammar. However, the Icelandic and Faroese languages are still quite similar to Old Norse and are therefore largely unintelligible to other Scandinavians.
The Finnish language, although currently spoken in a geographically Scandinavian country, is not technically a Scandinavian language. It is actually a member of the Finno-Ugric language family. Its closest relative is Hungarian.