Learn Icelandic: Ert þú tilbúinn fyrir ballið?

In Icelandic, adjectives must agree with the gender of the person they describe, even when using the pronoun þú. Because tilbúinn is in the masculine singular form, we know the speaker is asking a male. If they were asking a female, it would be Ert þú tilbúin?

I usually see this written as Ertu. Why are the words separated into Ert þú here?

Ertu is just a shortened, combined version of Ert þú that is extremely common in everyday speech and informal writing. Writing out the full Ert þú is perfectly correct but sounds slightly more formal or places a little extra emphasis on the word "you".

What case does the preposition fyrir trigger for the word ballið?

Here, fyrir takes the accusative case. When fyrir means "for" in the sense of preparation or an intended purpose, it is followed by the accusative. Because ball is a neuter noun, its accusative definite form (ballið) looks exactly identical to its nominative dictionary form.

Is tilbúinn related to the word búinn (finished)?

Yes! Búinn by itself usually means "finished" or "done", but when you add the prefix til-, the meaning changes. Tilbúinn means "ready" or "prepared".