Learn Icelandic: Við sjáum einhvern stóran hund úti.

Einhvern is the accusative masculine singular form of einhver. The verb sjá (to see) requires its direct object to be in the accusative case. Because the object here is hundur (a masculine noun), the word for "some" has to match its case and gender, becoming einhvern.

I remember the word for dog is hundur. Why is the -ur ending missing here?

Hundur is the dictionary form (nominative case), used when the dog is the subject of the sentence doing the action. Here, the dog is the object being seen, so it changes to the accusative case. For many masculine nouns ending in -ur, that ending simply drops off in the accusative, leaving just hund.

How does the adjective stóran fit into the grammar of this sentence?

Adjectives have to match the noun they describe in gender, number, and case. Because hund is masculine, singular, and in the accusative case, the basic adjective stór (big) takes the masculine accusative ending -an to become stóran.

Why does the sentence use úti for "outside" rather than út?