Learn Icelandic: Hún sér bilaðan bíl úr glugganum sínum.

Sínum is a form of the reflexive possessive pronoun sinn (one's own). You use it when the "owner" of the object is also the subject of the sentence. Since the subject hún (she) is looking out of her own window, we use sínum. If she were looking out of another woman's window, you would use hennar.

Why is sínum in the dative case here?

The pronoun has to match the gender, number, and case of the noun it belongs to, which is glugganum. The preposition úr (from/out of) always forces the following noun into the dative case. Because gluggi (window) is a masculine noun, the masculine, singular, dative form of sinn is sínum.

Why is it glugganum sínum with the definite article instead of just glugga sínum?

In Icelandic, when you place a possessive pronoun like sínum immediately after a noun, that noun usually needs to have the definite article (the "the" ending) attached to it. So glugganum literally means "the window", but together with sínum, the phrase simply translates to English as "her window".

Why is it bilaðan bíl instead of the dictionary forms bilaður bíll?