Tölvan og kennslubókin eru á skrifborðinu.

The endings -n and -in mark the definite form, like English “the”.

• tölva = a computer
  tölvan = the computer (feminine, weak declension → -an)

• kennslubók = a textbook
  kennslubókin = the textbook (feminine, strong declension → -in)

Icelandic usually attaches “the” to the end of the noun instead of using a separate word. So tölvan og kennslubókin literally means “computer-the and textbook-the” → the computer and the textbook.

eru is the 3rd person plural form of vera (to be).
Even though each noun is singular (tölvan = the computer, kennslubókin = the textbook), together they form a plural subject:

• tölvan → it is = hún er
• kennslubókin → it is = hún er
• tölvan og kennslubókin → they are = þær eru (and the verb form is just eru)

So:

• Tölvan er á skrifborðinu. – The computer is on the desk.
• Kennslubókin er á skrifborðinu. – The textbook is on the desk.
• Tölvan og kennslubókin eru á skrifborðinu. – The computer and the textbook are on the desk.

This is about prepositions and case.

• The preposition á (“on, in, at”) can take either accusative (movement to) or dative (location in/at/on).

Here we describe a static location: are on the desk, not onto the desk, so we use dative.

The noun skrifborð (desk, neuter) declines:

• nominative/accusative singular (indefinite): skrifborð
• dative singular (indefinite): skrifborði
• dative singular (definite): skrifborðinu

So:

• á skrifborði = on a desk (dative, indefinite)
• á skrifborðinu = on the desk (dative, definite)

Using á skrifborðið would be accusative and normally imply movement onto the desk (e.g. put it onto the desk), not just being located there.

-inu is the definite dative singular ending for many neuter nouns.

Breakdown:

• skrifborð = desk (neuter, indefinite)
• skrifborð-i = desk in dative (location: on/at the desk)
• skrifborð-inu = the desk in dative (“on the desk”)

So á skrifborðinu literally is on desk-the → on the desk.

Yes, that is also correct Icelandic.

• Tölvan og kennslubókin eru á skrifborðinu.
• Á skrifborðinu eru tölvan og kennslubókin.

Both mean “The computer and the textbook are on the desk.”

The second version (starting with Á skrifborðinu) puts extra emphasis on the location (on the desk), a bit like saying:

• On the desk are the computer and the textbook.

Icelandic word order is flexible, but the verb still comes early in the clause, so we keep eru right after the first element (Á skrifborðinu).

You would remove the definite endings and use the indefinite forms:

• Tölva og kennslubók eru á skrifborði.
  = A computer and a textbook are on a desk.

More literally:

• tölva = a computer
• kennslubók = a textbook
• á skrifborði = on a desk (dative, indefinite)

In natural Icelandic, speakers might often add extra context or quantify (e.g. Ein tölva og ein kennslubók… – one computer and one textbook), but the simple indefinite version above is grammatically correct.

Yes, they have grammatical gender, and it matters for endings and agreement.

• tölva – feminine
  
  • definite nom. sg.: tölvan
• kennslubók – feminine (compound of kennsla “teaching” + bók “book”)
  
  • definite nom. sg.: kennslubókin
• skrifborð – neuter
  
  • definite dat. sg.: skrifborðinu

Gender affects:

• which definite ending you use (-an, -in, -ið, -inu, etc.)
• which pronouns you’d use later:
  • tölvan / kennslubókin → hún (she/it, feminine)
  • skrifborðið / skrifborðinu → það (it, neuter)

In this particular sentence, gender mainly shows up in the different definite endings: -an, -in, -inu.

kennslubók is a compound noun:

• kennsla = teaching, instruction
• stem kennslu- (gen./combining form)
• bók = book

So kennslubók is literally a “teaching-book”, i.e. a textbook.

Icelandic makes lots of compound nouns this way:

• orðabók = orða (words) + bók (book) → dictionary
• skólabók = skóla (school) + bók (book) → schoolbook

Then you add the definite ending to the whole compound:

• kennslubók → kennslubókin (the textbook)

No. á is versatile and can mean several things depending on context:

• on (surface / location):
  • Bókin er á borðinu. – The book is on the table.
• in / at (locations, time expressions, institutions, etc.):
  • Hann er á Íslandi. – He is in Iceland.
  • Hún er á skrifstofu. – She is at the office.
• onto / to (with motion + accusative):
  • Hann setti bókina á borðið. – He put the book onto the table.

In Tölvan og kennslubókin eru á skrifborðinu, it’s clearly “on the desk” because it’s a static location.

Very rough English-style approximations (stressed syllable in CAPS):

• tölvan ≈ TURL-van

  • ö is like the vowel in British “bird” or German ö in schön.
  • stress on töl: TÖL-van

• kennslubókin ≈ KENNS-lu-boe-kin

  • kenn like “ken” but shorter, double n = short vowel.
  • slu like “sloo” but very short.
  • bói ≈ “boy” (Icelandic ó is a long “o” as in “go”).
  • stress on KENN: KENN-slu-bó-kin

• skrifborðinu ≈ SKRIF-borth-i-nu

  • skrif like “skriv”; i is short, as in “hit”.
  • borð: orð part is like “orth” but the ð is a soft th (often very weak or almost silent).
  • inu: “i-nu” with short vowels.
  • overall: SKRIF-bor-thi-nu with stress on skrif.

These are only approximations; Icelandic vowels are more precise and shorter/longer than English ones.