Mam kartę biblioteczną.

Mam is the 1st person singular form of the verb mieć (to have).

So:

• mam = I have
• masz = you have
• ma = he/she/it has

Polish usually builds the subject information into the verb ending, so one word can already mean I have.

In Polish, the subject pronoun is often omitted because the verb already shows who is doing the action.

• Mam kartę biblioteczną. = I have a library card.
• Ja mam kartę biblioteczną. also works, but ja is usually added only for emphasis, contrast, or clarification.

For example:

• Ja mam kartę, ale on nie ma. = I have a card, but he doesn’t.

So leaving out ja is completely normal.

Because mam takes a direct object, and in Polish the direct object often goes into the accusative case.

The dictionary form is:

• karta = card

But after mam (I have), it becomes:

• kartę = accusative singular

This happens because karta is a feminine noun. For many feminine nouns ending in -a, the accusative singular changes to -ę.

So:

• nominative: karta
• accusative: kartę

Because adjectives in Polish must agree with the noun they describe in:

• gender
• number
• case

Here, karta is:

• feminine
• singular
• accusative

So the adjective must match that.

Dictionary form:

• biblioteczna = feminine singular nominative

In this sentence:

• biblioteczną = feminine singular accusative

That is why both words change together:

• karta biblioteczna
  becomes
• kartę biblioteczną

It is the accusative singular.

Why?

• mam is followed by the thing possessed
• that thing functions as the direct object
• in Polish, direct objects are often in the accusative

So:

• kartę = accusative singular of karta
• biblioteczną = accusative singular feminine of biblioteczna

Yes. Polish word order is more flexible than English because case endings show the grammatical role.

The most neutral version is:

• Mam kartę biblioteczną.

But you may also hear:

• Kartę biblioteczną mam.
• Biblioteczną kartę mam.

These alternatives can sound more marked or emphatic. For a learner, the standard neutral order Mam kartę biblioteczną is the safest choice.

Biblioteczna is an adjective meaning library-related.

It comes from biblioteka = library.

So:

• karta biblioteczna = library card
• literally, something like library-related card

This is the normal Polish way to express the idea. Polish often uses an adjective where English uses a noun modifier.

The tricky part is the nasal vowels:

• ę
• ą

A simple learner-friendly guide:

• kartę sounds roughly like KAR-ten / KAR-teh depending on speed and accent
• biblioteczną sounds roughly like bee-blyo-TEHCH-non

A few pronunciation notes:

• bi in biblioteczną sounds like byi or bee depending on accent and how carefully you speak
• cz sounds like English ch in chop
• ą is a nasal vowel, often sounding a bit like on before some consonants
• ę is also nasal, but in everyday speech its exact sound can vary depending on what follows

You do not need to pronounce the nasal vowels perfectly right away to be understood.

Yes, very natural.

Mam kartę biblioteczną is a normal and idiomatic way to say that you have a library card. It sounds like standard everyday Polish.

A native speaker would readily understand it, and it does not sound overly formal or artificial.