Learn Icelandic: Fiskurinn er dauður.

In Icelandic, the definite article (the equivalent of 'the') is attached to the end of the noun. Fiskur is a masculine noun. To say 'the fish' as the subject of a sentence, you simply add the masculine suffix -inn to the base word, making it fiskurinn.

Why does the adjective dauður end in -ur here?

Adjectives in Icelandic must match the noun they describe in gender, number, and case. Since fiskur is a masculine, singular noun in the subject position, the adjective dauður (dead) takes the standard masculine singular ending -ur.

Could I use the word dáinn instead of dauður for a dead fish?

No, you generally wouldn't. In Icelandic, dauður is typically used when talking about dead animals, while dáinn is used respectfully for humans who have passed away.

How would the adjective change if we were talking about a feminine noun, like a mouse (mús)?

Because adjectives must match the gender of the noun, a feminine noun changes the adjective's ending. You would say músin er dauð (the mouse is dead), dropping the masculine -ur ending from the adjective.