Gestabókin er á borðinu.

Because Icelandic usually writes compound nouns as a single word.
So gestabók is one compound: gesta- + bók.

This is very common in Icelandic, and it often looks more like German-style compounding than English spelling.

Gesta- comes from gestur (guest), but in compounds Icelandic often uses a genitive form, very often the genitive plural.

So gestabók is historically something like book of guests, even though natural English says guestbook.

This does not mean there are multiple guestbooks. It is just the normal form used inside this compound.

That -in is the definite article, meaning the.

Icelandic usually puts the at the end of the noun instead of using a separate word in front:

• gestabók = guestbook / a guestbook
• gestabókin = the guestbook

So gestabókin already includes the.

Icelandic normally uses:

• a suffixed definite article for the
• no true indefinite article for a/an

So:

• bók can mean book or a book, depending on context
• bókin means the book

That is why the sentence does not need a separate word like English the before either noun.

It is in the nominative singular.

In this sentence, gestabókin is the subject, and subjects are normally in the nominative case.

Also, the gender is feminine, because the main noun bók (book) is feminine. In Icelandic compounds, the final part usually determines the gender and how the word declines.

Er means is.

It is the 3rd person singular present form of the verb vera (to be).

So:

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

Here the subject is singular, so er is the correct form.

Because á takes different cases depending on meaning:

• dative for a fixed location
• accusative for motion toward/onto something

Here the meaning is location: the guestbook is already there, on the table.
So Icelandic uses the dative:

• á borðinu = on the table

If you said á borðið, that would usually mean onto the table.

Borðinu is the dative singular definite form of borð (table).

The basic forms are:

• borð = table
• borðið = the table (nominative/accusative singular)
• borði = table in the dative
• borðinu = the table in the dative

So after á in a location meaning, you get á borðinu.

Not always. Its basic meaning is often on, but depending on context it can also be translated in ways like in or at.

In this sentence, the physical meaning on is the natural one:

• á borðinu = on the table

So it is best to learn á as a preposition whose exact English translation depends on context and case.

Yes. Gestabókin er á borðinu is a very normal neutral word order:

• subject: Gestabókin
• verb: er
• prepositional phrase: á borðinu

A native English speaker may notice that this looks quite familiar.

However, Icelandic also has verb-second behavior, so other word orders are possible for emphasis, for example:

• Á borðinu er gestabókin.

That still means the same thing, but the focus is different.

In Icelandic, the accent mark does not just show stress. It usually marks a different vowel sound.

So a and á are different vowels.

Á is pronounced roughly like the vowel in English now, though not exactly the same.

So the accent is an important part of the spelling, not an optional mark.