A könyvtárban nincs zene.

-ban/-ben is the inessive case, meaning in/inside something.
So könyvtár = library, könyvtárban = in the library.
It answers “where?” (Hol?): Hol? – A könyvtárban.

Hungarian uses vowel harmony.

• -ban is used after most “back-vowel” words.
• -ben is used after most “front-vowel” words.

Even though könyvtár contains ö (a front vowel), the last vowel á is a back vowel, so the ending is -ban: könyvtárban.

nincs means there isn’t / there are no and is the normal negative form of van in “existence/location” sentences.
You generally do not say nem van in standard Hungarian for this meaning.

• Van zene. = There is music.
• Nincs zene. = There is no music.

Hungarian doesn’t need a dummy subject like English there. Existence is expressed with van/nincs, and the thing that exists (or doesn’t) is stated directly:
Nincs zene. = (Literally) “Is-not music.”

Yes. a/az is the definite article (the).
A könyvtárban means in the library (a specific library understood from context).

If you mean “in a library (any library)”, Hungarian often still uses the definite article in such general statements, but you can also rephrase depending on context.

• a is used before a consonant sound: a könyvtár
• az is used before a vowel sound: az iskola

So here it’s A könyvtárban because könyvtár starts with k.

Because it’s not a direct object here. In Nincs zene, zene is the thing that does not exist (an “existential subject”), and it stays in the basic form (nominative), not accusative.

You’d use -t if it were a real object of an action verb:

• Nem hallok zenét. = I don’t hear music.

Yes, word order is flexible and changes emphasis. Both are correct:

• A könyvtárban nincs zene. = In the library, there is no music. (focus on the location)
• Nincs zene a könyvtárban. = There is no music in the library. (more neutral, or focus slightly more on “no music”)

Hungarian often puts what you’re contrasting/emphasizing right before the verb (nincs).

By default, A könyvtárban nincs zene usually means there is no music there / no music is present / no music is playing (practically: it’s quiet).
If you wanted to explicitly say “no music is playing,” you might also hear:

• Nem szól zene a könyvtárban. = No music is playing in the library.

Hungarian often doesn’t mark plural after numbers/quantities or in “there are” sentences unless it matters, but you can:

• A könyvtárban nincsenek dalok. = There are no songs in the library.

Here nincsenek is the plural form (for plural things).

Both exist:

• nincs is the most common everyday form.
• nincsen is a slightly longer variant (often more formal/explicit), same meaning.

So: A könyvtárban nincsen zene is also correct.

Approximate pronunciation notes:

• könyv has ö like German ö (or like the vowel in English learn for many speakers, but rounded).
• ny is like Spanish ñ (as in señor) or “n+y” said together.
• cs (not in this sentence, but common) is like ch; here you have nincs where cs = ch sound.

So nincs sounds roughly like neench (with a short i and cs = ch).