Miután ellenőriztem az adatokat és aláírtam a papírt, a pénztáros egy másolatot is adott.

Miután is a conjunction meaning after or after the fact that.

In this sentence, it introduces a time clause:

• Miután ellenőriztem az adatokat és aláírtam a papírt = After I checked the data and signed the paper

It is a single word, not mi után. As a conjunction, miután means after in the sense of after doing something.

Hungarian usually leaves out subject pronouns when they are clear from the verb ending.

So:

• ellenőriztem = I checked
• aláírtam = I signed

The I is already built into the verb form, so én is unnecessary unless you want extra emphasis.

The same happens with:

• adott = he/she gave

So a pénztáros adott already means the cashier gave without needing a separate pronoun.

Both are past tense, first person singular forms.

Very roughly:

• ellenőriz-tem = I checked
• aláír-tam = I signed

Hungarian past tense normally uses a -t- marker plus a personal ending.

So here:

• ellenőriz = check
• ellenőriztem = I checked

and

• aláír = sign
• aláírtam = I signed

The endings appear as -tam or -tem depending on vowel harmony and the shape of the verb stem.

A useful detail: in first person singular past, the definite and indefinite conjugations look the same, so -tam/-tem does not by itself show the difference.

Because they are direct objects, and Hungarian marks direct objects with the accusative ending -t.

So:

• az adatok = the data

• az adatokat = the data, as the thing being checked

• a papír = the paper

• a papírt = the paper, as the thing being signed

• egy másolat = a copy
• egy másolatot = a copy, as the thing being given

This -t on nouns is different from the -t that helps form the past tense on verbs.

Because a and az are the same definite article, just with two forms:

• az before a vowel sound
• a before a consonant sound

So:

• az adatokat because adatokat starts with a vowel
• a papírt because papírt starts with a consonant
• a pénztáros because pénztáros starts with a consonant

This is similar to English a/an, except in Hungarian it happens with the definite article.

Hungarian adat means data item, piece of information, or detail.

So:

• adat = one piece of data / one detail
• adatok = data / details
• az adatokat = the data / the details

In real-life situations like forms, registrations, or documents, Hungarian often uses adatok where English might say:

• the data
• the details
• the information

So the plural sounds very natural here.

This is about indefinite vs. definite conjugation.

Here the object is:

• egy másolatot = a copy

Because egy másolatot is indefinite, Hungarian uses the indefinite verb form:

• adott = gave

If the object were definite, you would expect the definite form instead:

• a másolatot adta = gave the copy

So:

• egy másolatot adott = gave a copy
• a másolatot adta = gave the copy

This is a very important Hungarian pattern.

Is means also or too.

Here it follows egy másolatot, so it is attached to that phrase:

• egy másolatot is adott = gave a copy too / also gave a copy

This usually suggests that the cashier gave something else as well, and a copy was one more thing.

Compare:

• A pénztáros is adott egy másolatot = The cashier also gave a copy
  Someone else gave one too, and the cashier did as well.

• Egy másolatot is adott a pénztáros = The cashier also gave a copy
  A copy was one additional thing that was given.

So the position of is matters.

Yes, Hungarian word order is flexible, and it often reflects topic, focus, and emphasis rather than fixed English-style structure.

This sentence is a very natural narrative order:

• Miután... = first the background action
• a pénztáros = then the subject/topic
• egy másolatot is adott = what happened next

So the sentence flows as:

• first I checked and signed
• then the cashier gave a copy too

Other word orders are possible, but they would change the emphasis. For example:

• Miután ellenőriztem az adatokat és aláírtam a papírt, egy másolatot is adott a pénztáros.
  More emphasis on a copy too

• A pénztáros egy másolatot is adott, miután ellenőriztem az adatokat és aláírtam a papírt.
  Same basic meaning, but the sentence starts from the main event instead

Because the sentence begins with a subordinate clause introduced by Miután:

• Miután ellenőriztem az adatokat és aláírtam a papírt, ...

That whole part is the after-clause, and it is separated from the main clause by a comma:

• ..., a pénztáros egy másolatot is adott.

This is standard Hungarian punctuation.

Yes. The verb is:

• aláír = to sign

Literally, it is related to write underneath, but in modern Hungarian it simply means sign.

So:

• aláírtam = I signed
• a papírt = the paper

Together:

• aláírtam a papírt = I signed the paper

That is the normal way to say it in Hungarian.