Amíg a fogorvos betömi az őrlőfogat, én nem beszélek.

Here amíg means while or as long as.

So the first clause means something like while the dentist is filling the molar.

A useful thing to know is that amíg can also mean until in other sentences. The exact meaning depends on context.

• Amíg dolgozom, nem zavarj. = While / As long as I’m working, don’t disturb me.
• Várj, amíg megjövök. = Wait until I come back.

In your sentence, the while / as long as meaning is the natural one.

Because Amíg a fogorvos betömi az őrlőfogat is a subordinate clause, and Hungarian normally separates subordinate clauses from the main clause with a comma.

So the structure is:

• Amíg a fogorvos betömi az őrlőfogat, = subordinate clause
• én nem beszélek. = main clause

This is very normal Hungarian punctuation.

Hungarian has two forms of the definite article:

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

So:

• a fogorvos because fogorvos starts with f
• az őrlőfogat because őrlőfogat starts with ő

It works a bit like choosing a vs an in English, although Hungarian uses the same spelling rule much more consistently.

Because the verb is using the definite conjugation.

Hungarian verbs often have two conjugation patterns:

• indefinite conjugation
• definite conjugation

You use the definite conjugation when the verb has a definite direct object.

Here the object is az őrlőfogat = the molar, which is definite, so the verb is:

• betömi = he/she fills it

Compare:

• A fogorvos betöm egy fogat. = The dentist fills a tooth.
  → indefinite object, so betöm
• A fogorvos betömi a fogat. = The dentist fills the tooth.
  → definite object, so betömi

Be- is a verbal prefix.

The base verb is töm / tömni, which has the basic idea of stuffing, packing, or filling something.

With be-, it becomes betömni, which often means fill in or fill up, and in a dental context it means to fill a tooth / cavity.

So:

• töm = fill, stuff
• betöm = fill in, fill up, plug up

In this sentence, betömi az őrlőfogat is the natural dental expression.

Because őrlőfogat is the accusative form.

The basic noun is:

• őrlőfog = molar

Since it is the direct object of the verb, it takes the accusative ending -t:

• őrlőfog → őrlőfogat

Hungarian often adds a linking vowel before -t when the noun ends in a consonant, so it is not just őrlőfogt, but őrlőfogat.

So here:

• az őrlőfogat = the molar as the object

Yes. Beszélek already tells you the subject is I, so én is not required.

Hungarian often leaves subject pronouns out unless there is a reason to include them.

When én is included, it usually adds emphasis or contrast. In this sentence it can feel like:

• As for me, I’m not talking
• I, on the other hand, am not talking

So:

• Amíg a fogorvos betömi az őrlőfogat, nem beszélek. = more neutral
• Amíg a fogorvos betömi az őrlőfogat, én nem beszélek. = more emphatic/contrastive

Because in a simple neutral Hungarian clause, nem usually comes directly before the finite verb.

So:

• beszélek = I speak / I am talking
• nem beszélek = I do not speak / I am not talking

That is the normal position for negation.

Beszélek is:

• 1st person singular
• present tense
• indefinite conjugation

It is present tense, but Hungarian present tense can cover several meanings depending on context:

• a general statement
• what is happening now
• sometimes even a near-future idea

Here it means something like I don’t talk / I’m not talking.

It is indefinite because there is no definite direct object after beszélek. The verb is being used intransitively, basically meaning talk or speak in general.

Yes, Hungarian word order is flexible, but changes in word order often change emphasis rather than basic meaning.

Your sentence has a very natural structure:

• Amíg a fogorvos betömi az őrlőfogat, én nem beszélek.

A few things are worth noticing:

• a fogorvos betömi az őrlőfogat is a neutral first clause
• én nem beszélek gives some emphasis to én

If you remove én, the sentence becomes more neutral:

• Amíg a fogorvos betömi az őrlőfogat, nem beszélek.

So yes, different word orders are possible, but the version you have is very normal and natural.