Ez egy egyházi iskola.

In Hungarian, the verb to be is usually left out in the present tense when the subject is third person and the sentence is something like X is Y.

So:

• Ez egy iskola. = This is a school.
• Ez egy egyházi iskola. = This is a church school / religious school.

But in other persons, the verb appears:

• Én tanár vagyok. = I am a teacher.
• Te diák vagy. = You are a student.

So the missing is here is completely normal.

Ez means this.

In this sentence, it stands alone as the subject:

• Ez = this
• egy egyházi iskola = a church school / a religious school

So the structure is basically:

• Ez
  • egy egyházi iskola
• This
  • (is) a church school

Here, egy is the indefinite article, meaning a or an.

So:

• egy iskola = a school
• egy egyházi iskola = a church school / a religious school

Hungarian uses egy much like English a/an in many cases.

Only the first egy is the article a/an.

The second word, egyházi, is one single word, not egy + something separate.

So:

• egy = a/an
• egyházi = church-related, ecclesiastical, religious

Even though egyházi starts with the letters egy, that part is not the article here.

Egyházi is an adjective meaning something like church, church-run, ecclesiastical, or sometimes religious, depending on context.

It comes from the noun egyház, which means church.

A very common Hungarian way to make an adjective from a noun is to add -i:

• egyház = church
• egyházi = church-related / ecclesiastical

So:

• egyházi iskola = literally church school
• in natural English: church school or religious school

Yes, this is the normal word order.

Hungarian adjectives usually come before the noun, just like in English:

• egyházi iskola = church school
• nagy ház = big house
• régi könyv = old book

So egyházi must come before iskola.

The whole sentence order is also very natural:

• Ez = this
• egy egyházi iskola = a church school

So the sentence is structured in a straightforward way.

Hungarian adjectives are much simpler than English speakers often expect if they know languages like French, German, or Spanish.

In Hungarian, adjectives do not change for:

• gender
• number
• usually not for the noun they describe in basic use

So:

• egyházi iskola = church school
• egyházi iskolák = church schools

The adjective egyházi stays the same.

Hungarian has no grammatical gender, so there is no masculine/feminine/neuter adjective agreement to worry about.

Normally, you would say Ez egy egyházi iskola.

Leaving out egy would sound unusual in ordinary neutral speech. Hungarian often uses the article in this kind of sentence when the predicate noun is indefinite:

• Ez egy iskola.
• Ez egy jó ötlet.
• Ez egy egyházi iskola.

Without egy, it might sound incomplete, unnatural, or stylistically special.

They mean different things.

• Ez egy egyházi iskola. = This is a church school.
• Ez az iskola = this school

In ez az iskola, ez directly modifies iskola, so Hungarian uses the article too:

• ez az iskola = this school
• literally something like this the school

But in Ez egy egyházi iskola, ez stands alone as the subject, and egy egyházi iskola is the predicate.

So the grammar is different.

A rough pronunciation guide:

• Ez ≈ ez
• egy ≈ something like edj (with a soft gy sound)
• egyházi ≈ edj-haa-zee
• iskola ≈ ish-ko-law

A few useful notes:

• Hungarian stress is usually on the first syllable of each word.
• gy is a soft consonant, somewhat like the d in British duke or the dy in module for some speakers, though not exactly the same.
• sz in Hungarian sounds like English s.
• A plain Hungarian s sounds like English sh.

So egyházi and iskola are pronounced quite regularly once you know the spelling rules.

The plural would be:

• Ezek egyházi iskolák. = These are church schools.

Breakdown:

• ezek = these
• egyházi = church / religious
• iskolák = schools

Notice that the adjective egyházi still does not change, and the verb are is still omitted in the present tense third person.