Sygnatura jest na karcie.

• sygnatura = a reference number / shelfmark / call number / identifying mark, depending on context
• jest = is
• na = on
• karcie = card in the locative singular form, so literally on the card

So the structure is very close to English: Sygnatura + jest + na karcie = The reference number is on the card.

Because na can take different cases depending on the meaning.

Here it expresses location: on the card.
When na means location, it normally takes the locative case.

• nominative: karta
• locative: karcie

So:

• na karcie = on the card
• not na karta, because that would use the wrong case

You may also see na kartę, but that is a different case and usually a different meaning, not simple location.

Because Polish nouns change form depending on their grammatical role.

Karta is a feminine noun. In the locative singular, many feminine nouns ending in -a change to -e or -ie.

So:

• karta → na karcie

The t becoming ci is a normal sound/spelling change in Polish in some forms. It is something you will see in other words too.

Jest is the 3rd person singular form of być (to be), so it means is.

In this sentence, it makes the statement complete and neutral:

• Sygnatura jest na karcie. = The reference number is on the card.

Could it be omitted? Sometimes Polish drops jest, especially in short notes, headlines, or very compressed style:

• Sygnatura na karcie.

But that sounds more like a label, note, or shorthand. For a normal full sentence, jest is the standard choice.

Yes. Polish word order is more flexible than English.

These are both possible:

• Sygnatura jest na karcie.
• Na karcie jest sygnatura.

Both are grammatical, but the emphasis changes a bit.

• Sygnatura jest na karcie focuses first on the reference number
• Na karcie jest sygnatura focuses first on where it is

English also does this a little, but Polish allows it more freely.

Because Polish has no articles.

English uses:

• a
• an
• the

Polish does not. So sygnatura can mean:

• a reference number
• the reference number

and na karcie can mean:

• on a card
• on the card

The exact meaning depends on context.

Yes. Sygnatura is feminine.

You can tell because it ends in -a, which is very common for feminine nouns in Polish.

That matters for agreement in other sentences, for example with adjectives:

• Ta sygnatura jest ważna. = This reference number is important.

In your sentence, jest does not change for gender in the present tense, so the gender is not very visible there.

A rough pronunciation guide:

• sygnatura ≈ syg-na-TOO-ra
• jest ≈ yest
• na ≈ nah
• karcie ≈ KAR-che

A more careful approximation:

• sy in sygnatura does not sound like English see
• je in jest sounds like ye
• cie in karcie sounds like a soft che

If you know IPA, roughly:

• sygnatura jest na karcie = /sɨɡnaˈtura jɛst na ˈkart͡ɕɛ/

Because Polish uses different prepositions depending on the relationship.

• na = on
• w = in

So na karcie means on the card, as in written on its surface.

If you said w karcie, that would mean in the card, which usually would not fit this context.

So if the reference number appears on the surface of a card, na karcie is the natural choice.

No, not usually.

• podpis = signature in the personal-handwriting sense
• sygnatura = reference number, designation, shelfmark, file number, identifying label

So even if English sometimes uses a similar-looking word, sygnatura is often not a person’s signature. It is more often an identifying code or mark.

That is an important vocabulary distinction for learners.

Yes, it is natural and correct.

It is a simple neutral sentence saying that the reference number is located on the card.

A native speaker would understand it immediately. The exact nuance of sygnatura depends on context, but the grammar of the sentence is straightforward and idiomatic.