Learn Icelandic: Þetta er bókin sem ég er að leita að. - This is the book that I am searching for.

They look identical but do completely different jobs! The first að is part of the continuous tense pattern vera að + infinitive, so er að leita translates to 'am searching'. The second að is the preposition 'for', because the verb leita requires að when you state what you are looking for.

Why is the second að pushed all the way to the end of the sentence?

This is called preposition stranding, and it works exactly like it does in English! Instead of saying 'the book for which I am searching', we say 'the book that I am searching for'. The relative word sem (that) connects the clauses, and the preposition að (for) gets left, or 'stranded', at the very end of the clause.

Bók is a feminine noun, so why does the sentence start with the neuter pronoun Þetta instead of Þessi?

When you introduce or point to something using the phrase 'This is...', Icelandic always uses the neuter pronoun Þetta, regardless of the noun's actual gender. You only use the feminine Þessi if it attaches directly to the noun as a modifier, as in Þessi bók er góð (This book is good).

The preposition að usually forces the following word into the dative case. Why is bókin in the nominative case here?