A csomagolás anyaga erős.

• a = the
• csomagolás = packaging, packing, or wrapping
• anyaga = its material / the material of it
• erős = strong

So the sentence is literally something like The packaging's material is strong, which in natural English can also be The material of the packaging is strong.

Because anyaga contains a possessive suffix.

• anyag = material
• anyaga = its material

In Hungarian, possession is usually shown on the possessed thing itself. So instead of using a separate word like of, Hungarian often adds a suffix:

• a csomagolás anyaga = the packaging's material / the material of the packaging

That final -a is the 3rd person singular possessive ending here, meaning something like its.

In Hungarian, the verb to be is often left out in the present tense, 3rd person singular when the predicate is a noun or adjective.

So:

• A csomagolás anyaga erős. = The material of the packaging is strong.

There is no written is, but the meaning is still there.

You do need the verb in other situations, for example:

• A csomagolás anyaga erős volt. = The material of the packaging was strong.
• Erős vagyok. = I am strong.

Hungarian usually uses one definite article for the whole noun phrase.

So:

• a csomagolás anyaga = the material of the packaging

You do not normally repeat the article before anyaga.

Also, a is the form used before consonant sounds. Before vowel sounds, Hungarian uses az.

Because Hungarian has an unmarked possessive construction that is very common.

So:

• a csomagolás anyaga = the packaging's material

Here csomagolás is the possessor, but it stays in its basic form.

Hungarian can also use a marked version with -nak/-nek:

• a csomagolásnak az anyaga

That version is possible, but it is usually more emphatic, more contrastive, or used in certain stylistic contexts. For a simple neutral sentence, a csomagolás anyaga is the normal choice.

Yes, basically.

Hungarian often expresses of relationships by using a possessive structure instead:

• a csomagolás anyaga = literally the packaging's material = naturally in English the material of the packaging

So if you are mentally looking for an English-style of, Hungarian often does it with possession instead.

Yes, you can also say Erős a csomagolás anyaga.

The version you were given:

• A csomagolás anyaga erős.

is a neutral statement: first the topic, then the new information.

If you say:

• Erős a csomagolás anyaga.

you put more emphasis on erős. It feels more like The material of the packaging is strong as opposed to weak, or as an important point.

So both are grammatical, but the word order changes the emphasis.

Because Hungarian adjectives do not behave like adjectives in many Indo-European languages.

A few important points:

• Hungarian has no grammatical gender
• adjectives do not change for masculine/feminine/neuter
• in this sentence, erős is just the basic adjective form

So:

• erős = strong

It stays the same here.

For comparison:

• erős csomagolás = strong packaging
• A csomagolás erős. = The packaging is strong.

In plural predicate use, you may see a plural ending:

• A dobozok erősek. = The boxes are strong.

But in your sentence, singular erős is exactly what you expect.

Not exactly.

• csomagolás = packaging / packing / wrapping
• csomagolóanyag = packaging material

So:

• A csomagolás anyaga erős. means The material of the packaging is strong.
• A csomagolóanyag erős. means The packaging material is strong.

The first talks about the material that a particular piece of packaging is made of. The second sounds more like you are talking about packaging material as a type of material.

A rough guide:

• A = short aw
• cs = ch as in church
• ny = like the ny in canyon
• s = sh
• á is a long vowel
• ő is a long rounded vowel that English does not really have

Very rough approximation:

aw CHO-ma-go-laash A-nya-ga EH-røøsh

A few useful details:

• Stress is always on the first syllable of each word in Hungarian.
• erős ends with sh, not s.
• csomagolás has the stress on cso.

Yes, it is natural and correct.

It sounds like a straightforward descriptive sentence. Depending on context, Hungarian speakers might also say something like:

• A csomagolás erős anyagból van. = The packaging is made of strong material.

But A csomagolás anyaga erős. is perfectly normal if you specifically want to talk about the material itself.