Learn Hardcore Icelandic: Ég er laus í dag.

Literally, í means “in” and dag is “day,” so í dag = “in (the) day.” Together they form the temporal expression “today.” Modern Icelandic spells it as two words (historically you might see ídag, but the current norm is í dag).

What case is dag in, and why does í take that case here?

Here dag is in the accusative singular (the basic form without -ur). The preposition í can govern either accusative or dative: with locations it’s dative (e.g. í skólanum “in the school”), but with time expressions it’s accusative, so í dag uses the accusative.

What does laus mean in this sentence?

Laus is an adjective meaning “free” or “available” (i.e. not busy). It can also mean “loose” in other contexts, but here it conveys “I have time today.”

Why is laus used instead of frí to say “I’m free”?

Frí is a noun meaning “a holiday” or “a break” (e.g. ég er frí can mean “I’m off work” or “I’m on vacation”). To express availability, you use the adjective laus. So ég er laus í dag = “I’m available today,” while ég er frí í dag = “I have the day off.”

Why isn’t there an article before laus, like English “a” or “the”?

Icelandic has no indefinite article, and predicative adjectives (those following vera “to be”) are used without any article. You simply say ég er laus, not ég er ein laus or anything similar.

How would I say “I’m free tomorrow” in Icelandic?

You’d say Ég er laus á morgun. Here á morgun (literally “on morning”) is the standard way to say “tomorrow.”

How do I ask someone “Are you free today?” in Icelandic?

Use the verb inversion for questions: Ertu laus í dag?

ertu = “are you” (2nd person singular of vera)
laus í dag = “free today”

How do I say “I’m not free today”?

Insert the negation ekki before the adjective: Ég er ekki laus í dag.

How is Ég er laus í dag pronounced, and why is Ég capitalized?

Pronunciation: Ég [jɛːɡ] (often the g is very soft or silent in fast speech), er [ɛːr], laus [lœiːs], í dag [iː ˈtaːɡ].
Ég is capitalized here because it’s the first word of the sentence. In Icelandic, the pronoun ég is only capitalized at the beginning of sentences (unlike English I which is always capital).