Az uszodában és a zuhanyzóban tilos futni.

Because Hungarian has two forms of the definite article:

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

So:

• uszoda starts with a vowel, so it becomes az uszoda
• zuhanyzó starts with a consonant, so it becomes a zuhanyzó

In this sentence, those nouns also take the ending -ban, so you get:

• az uszodában
• a zuhanyzóban

Here az is just the, not that.

The ending -ban / -ben means in or inside.

So:

• uszoda = swimming pool / pool facility

• uszodában = in the swimming pool / at the pool

• zuhanyzó = shower room / shower area
• zuhanyzóban = in the shower room / in the shower area

Hungarian often uses case endings like this instead of separate words such as in.

This is because of vowel harmony, which is a major feature of Hungarian.

Hungarian suffixes often have different forms depending on the vowels in the word:

• -ban is used after words with mostly back vowels
• -ben is used after words with mostly front vowels

Here:

• uszoda has back vowels, so: uszodában
• zuhanyzó also has back vowels, so: zuhanyzóban

A learner does not need to memorize every detail at once, but it is very useful to notice that Hungarian suffixes change shape to fit the word.

Hungarian normally repeats the article with each noun in a phrase like this, especially when each noun also has its own case ending.

So Az uszodában és a zuhanyzóban is the natural way to say:

• in the swimming pool and in the shower room

This is more like saying the full phrase twice, joined by és (and):

• az uszodában
• a zuhanyzóban

English often avoids repeating things, but Hungarian commonly keeps them.

Tilos means forbidden, not allowed, or prohibited.

In this sentence it works as the main predicate:

• tilos futni = running is forbidden / it is forbidden to run

This is a very common Hungarian pattern:

• Tilos dohányozni. = Smoking is forbidden.
• Tilos bemenni. = دخول is forbidden / It is forbidden to enter.

So tilos + infinitive is a standard way to express a rule or prohibition.

Because futni is the infinitive, meaning to run.

The ending -ni is the usual infinitive ending in Hungarian:

• fut = he/she/it runs, or the verb stem run
• futni = to run

After tilos, Hungarian uses the infinitive:

• tilos futni = to run is forbidden / running is forbidden

So this is not a full conjugated verb form. It is the dictionary-style to run form.

Yes, very close.

A very literal breakdown would be something like:

• Az uszodában és a zuhanyzóban = In the swimming pool and in the shower room
• tilos = forbidden
• futni = to run

So the whole sentence is structurally close to:

• In the swimming pool and in the shower room, to run is forbidden.

Natural English usually says:

• Running is forbidden in the swimming pool and shower room
• No running in the pool and shower area
• It is forbidden to run in the pool and shower area

Hungarian often leaves out the verb to be in the present tense when it would be is/are.

So where English says:

• It is forbidden

Hungarian can simply say:

• tilos

This is completely normal. Hungarian does not need a separate present-tense is here.

Usually uszoda means a swimming pool facility or pool building/area, not just the water.

If you specifically mean the pool basin or the water area, Hungarian often uses medence.

So:

• uszodában = at/in the swimming pool facility
• medencében = in the pool itself

In a warning sign like this, uszodában can refer more broadly to the pool area.

Because Hungarian word order is flexible, and putting the location first is very natural when the sentence is about a rule that applies in a certain place.

So:

• Az uszodában és a zuhanyzóban tilos futni.

naturally highlights where the rule applies first, and then states what is forbidden.

A rough English parallel would be:

• In the pool and in the shower area, running is forbidden.

That sounds a little formal in English, but it is normal in Hungarian.

És simply means and.

It connects the two location phrases:

• az uszodában = in the swimming pool
• a zuhanyzóban = in the shower room

So:

• Az uszodában és a zuhanyzóban = In the swimming pool and in the shower room