Ez az új dal nagyon szép.

This is a very common Hungarian pattern.

Hungarian often uses:

• ez a/az ... = this ...
• az a/az ... = that ...

So in Ez az új dal:

• ez = this
• az = the definite article the

The article changes form depending on the next word:

• a before a consonant
• az before a vowel

Because új starts with a vowel, Hungarian uses az:

• ez a dal = this song
• ez az új dal = this new song

So it may look like this the new song, but that is just how Hungarian normally builds this kind of phrase.

Here it is the.

That can be confusing, because az can also mean that in other sentences. But in this pattern, the first word tells you what is happening:

• ez az új dal = this new song
• az az új dal = that new song

So in your sentence:

• ez is the demonstrative: this
• az is the article: the

In Hungarian, the verb to be is often left out in the present tense when the subject is third person and the predicate is a noun or adjective.

So:

• A dal szép. = The song is beautiful.
• Ez az új dal nagyon szép. = This new song is very beautiful.

Hungarian does not need a separate word for is here.

But in other situations, the verb does appear, for example:

• Én boldog vagyok. = I am happy.
• Te fáradt vagy. = You are tired.
• A dal szép volt. = The song was beautiful.

So the missing is is normal Hungarian grammar, not an omission.

Because Hungarian adjectives normally come before the noun they describe, just like in English.

So:

• új dal = new song
• szép ház = beautiful house
• régi könyv = old book

That part is actually very familiar to English speakers.

Nagyon means very.

It is an adverb, and it comes before the adjective it modifies:

• nagyon szép = very beautiful
• nagyon jó = very good
• nagyon érdekes = very interesting

So the structure is:

• nagyon
  • adjective

That is why you get nagyon szép.

Hungarian has two forms of the definite article:

• a before a consonant
• az before a vowel

Since új begins with a vowel sound, the article must be az:

• a dal = the song
• az új dal = the new song

This is similar to how English uses a and an, although the Hungarian rule applies to the definite article instead.

Because dal is the subject of the sentence, so it stays in its basic dictionary form.

Hungarian adds endings when a noun has a special grammatical role, but the subject often has no ending.

Here:

• dal = song

Compare:

• Ez az új dal nagyon szép. = This new song is very beautiful.
  Here dal is the subject.

• Szeretem ezt az új dalt. = I like this new song.
  Here dalt is the object, so it gets -t.

So the plain form dal is exactly what you expect in this sentence.

No. Hungarian does not have grammatical gender like many European languages do.

That means there is no masculine/feminine difference in words like:

• ez
• az
• új
• dal
• szép

So Hungarian does not force you to choose between forms like this/he/she/masculine/feminine in the way some other languages do. That is one reason Hungarian grammar can feel simpler in this area.

A simple English-friendly approximation would be:

ez az ooy dal nawd-yon sape

A few key points:

• e in ez is like the e in bed
• a in az and dal is a more open sound, not the English ay
• új sounds roughly like ooy
• gy in nagyon is a soft sound; English does not have an exact match
• sz is pronounced like English s
• s by itself in Hungarian is pronounced like English sh
• é in szép is a long vowel, roughly like ay in careful speech, but without turning it into a diphthong too much

Also, Hungarian stress is normally on the first syllable of each word:

• EZ
• AZ
• ÚJ
• DAL
• NAgyon
• SZÉP

In Hungarian, accent marks usually show vowel quality and often vowel length. They are a real part of the spelling, not optional marks.

In this sentence:

• ú is a long vowel
• é is also a long vowel

So:

• u and ú are different
• e and é are different

This matters because vowel length can change the word itself. So when learning Hungarian, it is important to read and write the accents correctly.

Hungarian word order is more flexible than English word order, but the version you have is a very natural neutral sentence:

• Ez az új dal nagyon szép.

If you change the order, the meaning may stay similar, but the emphasis can change. For example:

• Nagyon szép ez az új dal.
  This puts more emphasis on very beautiful.

So yes, the word order can change, but the original sentence is a normal, straightforward way to say it.