Almenningsbókasafnið er opið.

Because Icelandic very often makes compound nouns by joining smaller words into a single word.

Here, the word is built roughly like this:

• almennings = of the public
• bóka = of books
• safn = collection
• -ið = the

So almenningsbókasafnið is literally something like the public book-collection, which is the normal Icelandic way to say the public library.

This kind of long compound is completely normal in Icelandic.

The ending -ið is the definite article, meaning the.

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

So:

• bókasafn = library
• bókasafnið = the library

In your sentence, the whole noun is definite: almenningsbókasafnið = the public library.

Because compounds often use a genitive form in their first part.

The noun almenningur means the public / common people / general public.
Its genitive singular form is almennings, meaning of the public.

So:

• almenningur = the public
• almennings- = public / of the public in a compound

That is why almenningsbókasafn literally means something like library of the public, i.e. public library.

Er means is.

It is the present singular form of the verb vera = to be.

So:

• vera = to be
• er = is

In this sentence, it links the subject and the adjective:

• Almenningsbókasafnið = the public library
• er = is
• opið = open

Because Icelandic adjectives must agree with the noun they describe in gender, number, and usually case.

The subject almenningsbókasafnið is neuter singular, so the adjective must also be neuter singular:

• opinn = masculine
• opin = feminine
• opið = neuter

Since almenningsbókasafnið is a neuter noun, the correct form is opið.

The gender of a compound noun usually comes from its last main element.

Here, the core noun is safn, which is a neuter noun.
Because bókasafn is built on safn, it is also neuter.
And because almenningsbókasafn is built on bókasafn, it stays neuter.

So the chain is:

• safn = neuter
• bókasafn = neuter
• almenningsbókasafn = neuter
• almenningsbókasafnið = still neuter, just definite

That is also why the adjective is opið.

It is in the nominative, because it is the subject of the sentence.

In a basic sentence like this:

• Almenningsbókasafnið = subject
• er = verb
• opið = predicate adjective

the subject normally appears in the nominative.

So this is the standard dictionary-like sentence pattern: subject + er + adjective.

You usually make a yes/no question by putting the verb first.

So:

• Almenningsbókasafnið er opið. = The public library is open.
• Er almenningsbókasafnið opið? = Is the public library open?

This verb-first pattern is very common in Icelandic questions.

The letter ð is usually pronounced like the th in this, not like the th in thing.

So in words like:

• bókasafnið
• opið

the final ð is a soft voiced th sound.

A few useful pronunciation points from this sentence:

• ó is roughly like o in go
• ð is like th in this
• stress normally falls on the first syllable of each word

So the strongest beats are on AL- in Almenningsbókasafnið and O- in opið.