A látogató az önkormányzatnál vár.

They are just two forms of the same definite article, the.

• a is used before a consonant sound: a látogató
• az is used before a vowel sound: az önkormányzatnál

So the choice is based on pronunciation, not on gender or case.

Látogató means visitor. It comes from the verb látogat, meaning to visit.

In this sentence it is clearly a noun, not an adjective, because it has the article a in front of it and functions as the subject: the visitor.

Hungarian usually expresses meanings like at, in, to, from, and so on with suffixes attached directly to the noun, not with separate prepositions.

So:

• önkormányzat = municipal government / local council / town hall office
• önkormányzatnál = at the municipal office / at the local council

The suffix -nál / -nél is called the adessive suffix, and it often means at or by.

Because of vowel harmony.

Many Hungarian suffixes have a back-vowel form and a front-vowel form:

• -nál
• -nél

The word önkormányzat contains back vowels such as o, á, a, so it takes the back-vowel form -nál.

This is something learners just get used to over time: the shape of the suffix depends on the vowels in the word it attaches to.

Because Hungarian does not usually use separate prepositions the way English does. Instead, it adds case endings to nouns.

So English:

• at the municipal office

becomes Hungarian:

• az önkormányzatnál

This is one of the biggest structural differences between English and Hungarian.

Because there is no direct object in the sentence.

Hungarian has two main verb conjugations:

• indefinite conjugation
• definite conjugation

Here, vár is the 3rd person singular indefinite form: he/she waits or is waiting.

You would use várja if there were a definite direct object, for example:

• A látogató a polgármestert várja. = The visitor is waiting for the mayor.

But in A látogató az önkormányzatnál vár, the sentence just says the visitor is waiting, without stating what or whom they are waiting for.

It can mean both.

Hungarian present tense often covers both:

• simple present: waits
• progressive present: is waiting

The exact meaning depends on context. In this sentence, English would very often translate it as is waiting, but grammatically Hungarian does not need a separate progressive form like English does.

Hungarian usually leaves subject pronouns out unless they are needed for emphasis or contrast.

The verb form itself already tells you the person and number:

• vár = he/she/it waits or is waiting

Since the subject is already stated as a látogató, there is no need to add a pronoun.

Hungarian word order is much more flexible than English word order, and it is strongly influenced by topic and focus.

Here:

• A látogató is the topic: what the sentence is about
• az önkormányzatnál vár says something about that topic

Putting the place phrase before the verb is very natural in Hungarian. It presents waiting at the municipal office as the main statement about the visitor.

A different order can also be possible, but it may sound different in emphasis or rhythm.

Not exactly.

• az önkormányzatnál = at the municipal office / at the local council
• az önkormányzatban = in the municipal office / inside the municipality building/administration

With institutions, Hungarian often prefers -nál / -nél when the meaning is something like at the office, at the bank, at the doctor, at city hall, and so on.

So in this sentence, önkormányzatnál is the more natural choice if the idea is simply that the visitor is there waiting at that institution.