Learn Icelandic: Búðin er opin á laugardögum. - The shop is open on Saturdays.

Because the shop is open on Saturdays in general (a recurring event). In Icelandic, when you use á to say something happens regularly on a day of the week, the day takes the dative case. If you were talking about just one specific Saturday coming up, you would use the accusative case.

How does the word laugardagur turn into laugardögum?

The preposition á for recurring days requires the plural dative form. Almost all plural dative nouns in Icelandic end in -um. When we add -um to a word, an a earlier in the word often changes to an ö (this is a common rule called the U-shift). This is why the dagur part turns into dögum, giving us laugardögum.

Why is the word for open spelled opin and not opinn?

Adjectives in Icelandic must match the gender of the noun they describe. The word for shop, búð, is a feminine noun. Therefore, the adjective drops the extra n from its masculine dictionary form (opinn) to become the feminine form opin.

Where is the word for "the" in this sentence?

Icelandic usually doesn't use a separate word for "the". Instead, the definite article is attached directly to the end of the noun. The base feminine noun is búð (a shop), and adding the suffix -in gives us búðin (the shop).