Learn Icelandic: Þetta er bók og bókin er þar. - This is a book and the book is there.

In Icelandic, we don't have a separate stand-alone word for "the". Instead, we add a suffix to the end of the word. Bók means "book" or "a book" (indefinite). When we talk about that specific book again, we add the definite suffix to make bókin ("the book").

How do I know to use -in at the end of bókin instead of -inn or -ið?

The "the" suffix you use depends on the noun's gender. Bók is a feminine noun, and the definite suffix for feminine nouns in the subject position is -in. If the word were masculine, you would use -inn, and if it were neuter, you would use -ið.

Since bók is feminine, why do we use the neuter word þetta for "this"?

When introducing something by saying "This is a..." or "That is a...", Icelandic always uses the fixed neuter pointing word þetta, no matter the gender of the noun that comes after it! You only need to match the noun's gender if you are attaching "this" directly to the noun, like saying "this book" rather than "This is a book".

Is there a missing word for "a" before bók?

No, Icelandic actually doesn't have a word for "a" or "an" at all. The bare word bók by itself naturally translates to "a book" when we introduce it for the first time in a sentence like this.