Learn Icelandic: Mig langar að fara til Danmerkur.

The verb langa (to want) is an impersonal verb. Instead of saying 'I want', Icelandic expresses it more like 'it wants me'. Because of this, the word for I is forced into the accusative case, becoming Mig.

What is the purpose of að before fara?

Here, að works just like the English word 'to' before an action. When you say you want 'to go', you use að followed by the infinitive (the basic dictionary form) of the verb, which is fara.

Why is the word for Denmark written as Danmerkur instead of Danmörk?

This happens because of the preposition til (to). In Icelandic, til always forces the noun that follows it into the genitive case, which is often used to show possession or, as in this lesson, destinations with til. The genitive form of Danmörk is Danmerkur.

Why did the vowel ö in Danmörk change to an e in Danmerkur?

When this noun changes to the genitive case, it adds -ur at the end. In Icelandic, changing the ending of a word sometimes causes the vowels inside the word to shift. Here, the ö shifts to an e. You will see this exact same pattern in any other word ending in -mörk!