Bókin liggur á gólfinu við rúmið.

In Icelandic, the definite article is usually attached to the end of the noun.

• bók = book
• bókin = the book

So -in here is the suffixed definite article for this form of a feminine noun.

Liggur is the present tense, 3rd person singular form of the verb liggja, which means to lie.

It matches the singular subject bókin.

So:

• ég ligg = I lie
• þú liggur = you lie
• hún/hann/það liggur = she/he/it lies

Here, the book is grammatically singular, so liggur is the correct form.

Icelandic often prefers a position verb where English might simply use is.

So instead of saying the equivalent of The book is on the floor, Icelandic very naturally says The book lies on the floor:

• liggja = lie
• sitja = sit
• standa = stand
• hanga = hang

This is very common in Icelandic.
You can sometimes use er, but liggur is more natural here because a book is understood as resting in a lying position.

Because á changes case depending on whether it describes:

• location = dative
• movement toward/onto something = accusative

In this sentence, the book is already there. It is lying on the floor, so this is location, not movement.

That is why we get:

• á gólfinu = on the floor (dative)

Compare:

• Bókin liggur á gólfinu. = The book is lying on the floor.
• Hún setur bókina á gólfið. = She puts the book onto the floor.

So gólfinu is the dative singular definite form of gólf.

No, not here.

In this sentence, við means something like:

• by
• next to
• beside

So við rúmið means by the bed or next to the bed.

This is a very important point, because við does not always mean English with.

Because the preposition við normally takes the accusative case.

The noun is rúm = bed, and here we have:

• rúmið = the bed (accusative singular definite; same form as nominative singular definite for this noun)

So:

• við rúm = by a bed
• við rúmið = by the bed

Even though the form looks simple, the important grammar point is that við governs the accusative.

Because Icelandic nouns change form for case, and the definite article is built into those forms.

Here the two nouns are in different cases:

• gólfinu = dative singular definite
• rúmið = accusative singular definite

So both mean the floor / the bed, but they appear in different grammatical roles after different prepositions:

• á
  • dative for location → á gólfinu
• við
  • accusative → við rúmið

This is one of the biggest differences from English: Icelandic often shows grammar through noun endings.

This is a very normal, straightforward Icelandic word order:

• Bókin = subject
• liggur = verb
• á gólfinu við rúmið = location

So it is basically:

The book lies on the floor by the bed.

However, Icelandic word order is more flexible than English, especially for emphasis. For example, a location phrase could be moved to the front:

• Á gólfinu við rúmið liggur bókin.

That sounds more marked or literary, but it is grammatical. In main clauses, Icelandic still usually keeps the finite verb in second position.

In meaning, it helps describe the location of the book as a whole.

So the sense is:

• the book is on the floor,
• and that floor-location is by the bed.

In natural English, we would also usually understand it as one combined location:

on the floor by the bed

So you do not need to force a strict distinction here. It is best understood as the full place where the book is lying.

They are:

• bók = book
• liggja = to lie
• gólf = floor
• rúm = bed

And the forms in the sentence are:

• bókin = the book
• liggur = lies / is lying
• gólfinu = the floor
• rúmið = the bed

This is useful because Icelandic dictionaries usually list the basic form, not the exact inflected form you see in the sentence.