Ég tek bókina niður af hillunni.

It can mean either, depending on context. Icelandic simple present often covers both:

• I take
• I am taking

So Ég tek bókina niður af hillunni can describe a general action or something happening right now. Context tells you which reading is intended.

taka is the infinitive, meaning to take.

tek is the 1st person singular present form, so it means I take / I am taking.

This verb is a little irregular:

• að taka = to take
• ég tek = I take
• þú tekur = you take
• hann/hún/það tekur = he/she/it takes

So after ég, you need tek, not the infinitive taka.

Because bókina is the form that means the book as a direct object.

Here is the difference:

• bók = book / a book
• bókin = the book as a subject form
• bókina = the book as an object form

The verb taka normally takes the accusative case for its direct object, so the book becomes bókina.

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

So:

• bókina = the book
• hillunni = the shelf

That is why you do not see a separate word like English the before the nouns.

niður means down.

In this sentence, it works like a particle/adverb and gives the sense of take down, not just take. It shows downward movement from a higher place.

So the combination taka ... niður is very similar to English take ... down.

Yes, you could say Ég tek bókina af hillunni, and it would still make sense.

But niður makes the action more precise. It tells you that the book is being taken down from the shelf, not just taken away from it in some less specific way.

With a shelf, niður sounds very natural because shelves are usually above hand level or at least treated as a place something is taken down from.

Because af takes the dative case.

The noun is hilla = shelf. After af, it changes form:

• hilla = shelf
• hillu = shelf in dative, indefinite
• hillunni = the shelf in dative

So af hillunni means from/off the shelf.

The ending changes because Icelandic nouns change form depending on their role in the sentence and the preposition used with them.

Because Icelandic distinguishes between off/from a surface and out of an enclosed space.

• af = off/from a surface or support
• úr = out of the inside of something

A book sits on a shelf, not inside it, so Icelandic uses af:

• af hillunni = off/from the shelf

But you would use úr for something like:

• úr kassanum = out of the box
• úr töskunni = out of the bag

You could say that too. Both orders are possible.

• Ég tek bókina niður af hillunni
• Ég tek niður bókina af hillunni

The sentence you were given is completely natural. Icelandic often allows this kind of variation with particles like niður. The meaning stays basically the same, though the rhythm or emphasis can shift slightly.