Bókin er efst á hillunni.

Because bókin means the book, while bók means a book or just book.

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word like English the.

• bók = book
• bókin = the book

Here, bókin is also in the nominative singular, because it is the subject of the sentence.

Er is the present-tense form of vera, which means to be.

So:

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

In this sentence, Bókin er... means The book is...

Efst means topmost, highest, or at the top.

In this sentence, it describes the book’s position. It tells you that the book is at the highest point relative to the shelf.

A good natural English equivalent is often:

• at the top
• topmost
• highest up

Here, efst is best understood as an adverbial/predicative form describing position, not as an adjective directly modifying a noun.

That matters because it does not change to match bókin in gender, number, or case.

Compare:

• Bókin er efst á hillunni. = The book is at the top / topmost on the shelf.
• á efstu hillunni = on the top shelf

In á efstu hillunni, efstu is an adjective describing hillunni. In the original sentence, efst stands on its own.

Because á can take different cases depending on whether you mean location or motion.

• dative = location, being somewhere
• accusative = motion toward something

Here the book is already located there, so Icelandic uses the dative:

• á hillunni = on the shelf

If you were talking about moving the book onto the shelf, you would use the accusative:

• Settu bókina á hilluna. = Put the book onto the shelf.

This is a very common Icelandic pattern.

The dictionary form is hilla = shelf.

In the sentence, it appears as hillunni because it is:

• singular
• definite = the shelf
• dative after á for location

So the form changes from hilla to hillunni.

You do not need to think of it as a simple one-piece ending that always works the same way for every noun, but for this noun the important pattern is:

• hilla = a shelf
• hillunni = on the shelf / the shelf (dative singular definite)

• Bókin is nominative singular
• hillunni is dative singular

Why?

• Bókin is the subject, so it is nominative.
• hillunni comes after á in a location meaning, so it is dative.

This is a useful sentence for seeing how Icelandic marks grammatical roles with case endings.

Because Icelandic normally expresses definiteness by attaching the article to the noun.

So instead of:

• English: the book
• Icelandic: bókin

And instead of:

• English: the shelf
• Icelandic: hillan / hillunni depending on case

There is a separate article in Icelandic (hinn, hin, hið), but in ordinary sentences like this, the suffixed article is the normal choice.

Not really. They do different jobs.

• efst tells you the position: at the top
• á hillunni tells you where that position is: on the shelf

So together they mean something like:

• the book is at the top on the shelf

Without á hillunni, efst would be less specific. Without efst, you would only know that the book is on the shelf, not where on it.

No, the meaning is related but not identical.

• Bókin er efst á hillunni.
  The book is at the top on the shelf / topmost on the shelf.

• Bókin er á efstu hillunni.
  The book is on the top shelf.

In the first sentence, efst describes the book’s position.
In the second, efstu describes which shelf it is on.

That distinction is very useful in Icelandic.

This is a normal, neutral Icelandic sentence order:

• Bókin = subject
• er = verb
• efst á hillunni = complement describing location/position

So it follows a pattern similar to English:

• The book
  • is
    • at the top on the shelf

Icelandic word order can change for emphasis, but this version is the most straightforward and natural basic order.

Very simply:

Negative:

• Bókin er ekki efst á hillunni.
• The book is not at the top on the shelf.

Yes/no question:

• Er bókin efst á hillunni?
• Is the book at the top on the shelf?

Notice that in the question, the verb er comes first. That is a common Icelandic pattern, just like in English Is the book... ?