Learn Icelandic: Þetta er ný bók.

When you introduce something by saying "This is a...", Icelandic always uses the neuter form Þetta, no matter the gender of the noun that follows. You only match the gender (like using the feminine Þessi) if you were saying "This book is..." (Þessi bók er...).

Why is the word for "new" spelled ný here instead of nýr?

Adjectives in Icelandic change their endings to match the gender of the noun they describe. Because bók is a feminine noun, the adjective must be in its feminine form. The standard dictionary form (which is masculine) is nýr, but the feminine singular form drops the R to become ný. If the noun were neuter, you would use nýtt.

Where is the word for "a" in this sentence?

Icelandic doesn't have an indefinite article (the words "a" or "an" in English). You simply say ný bók, which translates word-for-word as "new book" but means "a new book" in English.