Skógurinn er jafn rólegur og bókasafnið.

Because Icelandic usually puts the definite article on the end of the noun.

• skógur = forest
• skógurinn = the forest

So -inn here is the masculine singular nominative definite ending. This is one of the first big differences from English: Icelandic often says forest-the where English says the forest.

For the same reason: the noun is definite.

• bókasafn = library
• bókasafnið = the library

Here the ending -ið marks the with a neuter singular nominative noun. So both nouns in the sentence are definite: the forest and the library.

jafn ... og ... means as ... as ...

So:

• jafn rólegur = as quiet / as calm
• og bókasafnið = as the library

Together, jafn rólegur og bókasafnið means as quiet as the library.

This is a very common Icelandic comparison pattern:

• Hann er jafn hár og ég. = He is as tall as I am.
• Bíllinn er jafn gamall og húsið. = The car is as old as the house.

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

The subject here is skógurinn, which is:

• masculine
• singular
• nominative

So the adjective must match that, giving rólegur.

Compare:

• skógurinn er rólegur = masculine singular
• borgin er róleg = feminine singular
• barnið er rólegt = neuter singular

English adjectives do not change like this, but Icelandic ones often do.

Because the adjective in the actual sentence agrees with skógurinn, not with bókasafnið.

A useful way to understand it is to expand the sentence:

Skógurinn er jafn rólegur og bókasafnið er rólegt.

In the shortened version, the second er rólegt is left out because it is understood. So:

• rólegur matches skógurinn
• the hidden rólegt would match bókasafnið

This is very similar to English ellipsis in sentences like The forest is as quiet as the library (is).

Both nouns are in the nominative singular in this sentence.

• skógurinn is nominative because it is the subject of the sentence.
• bókasafnið is also nominative in the comparison, because the full underlying structure is basically: Skógurinn er jafn rólegur og bókasafnið er rólegt.

Since bókasafnið is understood as the subject of the second clause, nominative makes sense there too.

Because Icelandic is generally a verb-second language in main clauses. In a normal statement, the finite verb often comes early, typically in second position.

Here the structure is straightforward:

• Skógurinn = subject
• er = is
• jafn rólegur og bókasafnið = complement/comparison

So Skógurinn er ... is the expected order.

It can mean either, depending on context.

For a place like a forest or a library, rólegur often suggests:

• quiet
• calm
• peaceful

So in this sentence, the idea is that the forest is as calm/quiet as the library. English may choose different words depending on style, but Icelandic rólegur covers that general peaceful, not noisy feeling.

A rough English-friendly guide would be:

• Skógurinn ≈ SKOH-guh-rin
• er ≈ ehr
• jafn ≈ yapn
• rólegur ≈ ROH-leh-gur
• og ≈ okh or ogh depending on speech
• bókasafnið ≈ BOH-ka-sapn-ith

A few helpful pronunciation notes:

• j in Icelandic sounds like English y
• ó is a long vowel, roughly like oh
• ð is related to the th sound, though in endings it can be weaker than English speakers expect
• fn in safnið is pronounced quite tightly, not as two fully separate syllables

It would change to match the gender of the new subject.

For example:

• Borgin er jafn róleg og bókasafnið.
  The city is as quiet as the library.
  Here borgin is feminine, so the adjective is róleg.

• Barnið er jafn rólegt og bókasafnið.
  The child is as quiet as the library.
  Here barnið is neuter, so the adjective is rólegt.

So the pattern stays the same, but the adjective form changes to agree with the subject.