Klub koristi biblioteku.

Because biblioteku is in the accusative case, which is used for the direct object of the verb.

• Basic form: biblioteka = library
• In the sentence, the library is the thing being used, so it becomes biblioteku

This is a very common pattern with feminine nouns ending in -a:

• nominative: biblioteka
• accusative: biblioteku

So Klub koristi biblioteku literally has:

• Klub = subject
• koristi = uses
• biblioteku = direct object

Because klub is the subject of the sentence, so it is in the nominative case, which is the dictionary form.

In this sentence:

• klub = the club
• it is doing the action
• so it stays klub

Also, klub is a masculine noun, and many masculine inanimate nouns look the same in nominative and accusative singular. But here it is nominative anyway, because it is the subject.

Koristi is the 3rd person singular present tense of the verb koristiti, meaning to use.

So:

• ja koristim = I use
• ti koristiš = you use
• on/ona/ono koristi = he/she/it uses

Since klub is grammatically singular, the verb must also be singular:

• Klub koristi = The club uses

No. Serbian does not use articles like English the or a/an.

So Klub koristi biblioteku could mean:

• The club uses the library
• A club uses a library
• The club is using the library

The exact meaning depends on context.

That is why Serbian learners of English have to learn articles, and English speakers learning Serbian have to get used to not seeing them.

Yes, Serbian word order is more flexible than English word order because cases show what each noun is doing.

The neutral order here is:

• Klub koristi biblioteku.

But you may also hear:

• Biblioteku koristi klub.
• Koristi biblioteku klub.

These can sound more marked or emphasize different parts of the sentence.

Even if the order changes, biblioteku still stays the object because its accusative ending -u shows its role.

In Serbian, case endings often tell you that.

Here:

• klub is in nominative form, which is the normal subject form
• biblioteku has the accusative ending -u, which shows it is the direct object

So even without relying only on word order, Serbian grammar tells you:

• the club is doing the action
• the library is receiving the action

That is one of the big differences from English.

Koristiti is usually imperfective.

That means it describes:

• repeated action
• habitual action
• ongoing action
• general fact

So Klub koristi biblioteku can mean something like:

• The club uses the library
• The club is using the library
• The club makes use of the library

If you wanted a perfective idea, Serbian would normally use a different verb or a different construction, depending on the exact meaning.

Yes, gender matters a lot in Serbian.

In this sentence:

• klub is masculine
• biblioteka is feminine

Gender matters because it affects:

• case endings
• adjective forms
• pronouns
• sometimes past tense agreement

For example:

• dobar klub = good club
• dobra biblioteka = good library

And in accusative:

• dobru biblioteku = good library

So learning the gender of each noun is very important.

A rough English-friendly pronunciation is:

• Klub ≈ kloop
• koristi ≈ ko-REE-stee
• biblioteku ≈ bee-blee-oh-TEH-koo

A few helpful notes:

• Serbian r is rolled or tapped
• stress is not usually marked in normal writing
• each letter is pronounced quite consistently

So the whole sentence sounds roughly like:

• kloop ko-REE-stee bee-blee-oh-TEH-koo

Klub is a broad word, much like English club. It can refer to:

• a social club
• a sports club
• an organization or association
• sometimes a nightclub, depending on context

So without more context, Klub koristi biblioteku could mean:

• a student club uses the library
• a sports club uses the library
• some other organization uses the library

Context tells you which kind of club is meant.