Learn Hardcore Icelandic: Bækurnar eru á hillunni fyrir ofan rúmið. - The books are on the shelf above the bed.

Bækur is the indefinite plural form of bók (book) → bækur = books.
Bækurnar adds the definite ending -nar, so it means the books.

Icelandic doesn’t use a separate word like the; instead, the definite article is a suffix:

singular: bók → bókin = the book
plural: bækur → bækurnar = the books

There’s also a vowel change (ó → æ) when going from bók to bækur/bækurnar. That’s a regular sound change pattern in many Icelandic noun plurals.

Why is it eru and not er?

Both are forms of vera (to be):

er = is / am (3rd person singular, also 1st singular)
eru = are (3rd person plural)

Since the subject Bækurnar is plural (the books), the verb must also be plural: Bækurnar eru… = The books are…

What does á hillunni mean, and why is hillunni in that form?

á hillunni = on the shelf.

á means on (among other things).
The basic word for shelf is hilla.
Here we have hillu + inni → hillunni.

Grammatically:

á with a static location (on something, not moving to it) takes the dative case.
hilla is a feminine noun; dative singular definite is hillunni.

Rough pattern for a feminine -a noun like hilla:

nominative sg.: hilla (a shelf)
dative sg.: hillu
dative sg. definite: hillunni (on the shelf)

So á (location) → dative → hillunni.

Why do we need both á hillunni and fyrir ofan rúmið? Aren’t they saying the same thing?

They describe two different relationships:

á hillunni = where the books are → on the shelf
fyrir ofan rúmið = where the shelf is (and therefore the books) → above the bed

Put together, the sentence means: the books are on the shelf, and that shelf is above the bed.

What does fyrir ofan mean exactly, and why is it two words?

fyrir ofan functions as a single preposition meaning above / over (in a vertical sense).

It’s written as two separate words, but you should learn it as one unit:

fyrir ofan + accusative → above [something]
- fyrir ofan rúmið = above the bed

So you can think of fyrir ofan as the Icelandic equivalent of English “above”.

Why is it rúmið and not some other form of rúm?

The base word rúm means bed or space. It is a neuter noun.

Neuter nominative/accusative singular is just rúm.
The definite form (the bed) in nominative/accusative singular is rúmið (rúm + ið).

fyrir ofan governs the accusative case, so we need the accusative definite form:

accusative singular definite: rúmið = the bed

So:

rúm = a bed
rúmið = the bed

Why is hillunni in the dative, but rúmið in the accusative?

Different prepositions require different cases:

á with location (on, at, in) normally takes dative
- á hillunni → hilla in dative singular definite = hillunni
fyrir ofan takes accusative, even when it’s about a static location
- fyrir ofan rúmið → rúm in accusative singular definite = rúmið

So the case is decided not by the verb, but by the preposition in each phrase.

Could I say the sentence in a different word order, like Bækurnar eru fyrir ofan rúmið á hillunni?

That word order is grammatically possible, but it sounds awkward and less natural. Native Icelandic tends to keep these prepositional phrases in a more “inside-out” order:

Immediate location of the subject: á hillunni (on the shelf)
Then a wider reference point: fyrir ofan rúmið (above the bed)

So Bækurnar eru á hillunni fyrir ofan rúmið is the natural order. Reordering often changes emphasis or just sounds clumsy.

How does á work in general? Sometimes it seems to mean on, but sometimes in or at.

á is a very common preposition with several meanings, depending on context:

on a surface: á borðinu = on the table
in / at a place: á Íslandi = in Iceland; á skrifstofunni = at the office
with motion (onto): setja bókina á borðið = put the book on(to) the table (accusative)

Case use:

á + dative → usually location (on, in, at)
á + accusative → usually direction / movement (onto, to)

In your sentence, á hillunni is location → dative.

What are the genders of bók/bækur, hilla, and rúm, and why do they matter?

bók (pl. bækur) = feminine
hilla = feminine
rúm = neuter

Gender matters because it affects:

Which endings nouns take in different cases and numbers.
The form of adjectives and pronouns that agree with them.
The shape of the definite article suffix.

Examples:

fem. plural definite: bækur → bækurnar
fem. dat. sg. definite: hilla → hillunni
neut. acc. sg. definite: rúm → rúmið

How would I say “The books are above the bed” without mentioning the shelf?

You can just omit the shelf phrase:

Bækurnar eru fyrir ofan rúmið.
= The books are above the bed.

Now fyrir ofan rúmið directly describes the position of the books instead of the shelf.

How do I pronounce the special letters in Bækurnar eru á hillunni fyrir ofan rúmið?

Key points for an English speaker:

æ in Bækurnar: like the i in eye or my → [ai].
á in á and rúmið’s ú:
- á is like ow in cow → [au].
- ú is like oo in food but usually shorter.
ll in hillunni: in standard pronunciation it’s often like tl or tl̥; roughly HITL-oonni (very approximate).
ð in rúmið: a soft voiced th sound (as in this), but word-finally it’s often very weak or almost silent in fluent speech.

You don’t need to get it perfect at first, but knowing that æ ≈ eye, á ≈ ow, and ð ≈ th is a good start.