Learn Hardcore Icelandic: Sólin skín í dag.

Sólin is the definite form, meaning “the sun.” In Icelandic, the definite article is a suffix rather than a separate word. The base noun sól means “sun,” and adding -in (the feminine singular article) gives sólin = “the sun.”

What is the infinitive of skín, and what tense/person is this form?

The infinitive is skína (to shine). The form skín is present indicative, 3rd person singular — he/she/it shines.

Why is í dag used for “today” instead of dagur or dagurinn?

í dag is a fixed time‐expression meaning “today.”

dagur is the bare noun “day” (nominative singular).
dagurinn is “the day” (definite nominative).
When telling time with í, you use the accusative form dag without the article. If you wanted “during the day,” you would say í daginn (accusative definite).

What case is dag in í dag?