A társadalomnak vannak szabályai.

Because Hungarian often shows the possessor with -nak / -nek in sentences meaning someone has something.

The basic pattern is:

X-nek van Y-ja = X has Y

So:

• a társadalomnak = to society / for society grammatically, but here it marks society as the possessor
• vannak szabályai = there are its rules

So the whole sentence literally works like:

To society, there are rules of it.

That is the normal Hungarian way to say Society has rules.

Hungarian normally does not use a separate verb like English have for simple possession.

Instead, it uses van / vannak, which mean is / are.

So possession is expressed with a structure like:

• A lánynak van könyve. = The girl has a book.
• A lánynak vannak könyvei. = The girl has books.

In your sentence:

• vannak is used because szabályai is plural

So Hungarian says possession more like:

For society, there are rules.

Because the thing possessed is plural.

Compare:

• A társadalomnak van szabálya. = Society has a rule.
• A társadalomnak vannak szabályai. = Society has rules.

So in this pattern, van / vannak agrees with the possessed thing:

• van = singular
• vannak = plural

It does not agree with társadalom, which is singular.

Because Hungarian marks possession on the possessed noun itself.

• szabályok = rules
• szabályai = his/her/its rules or the rules belonging to it

Since society is the possessor, Hungarian adds a possessive ending to szabály.

So:

• szabályok = just rules
• szabályai = its rules

This is very common in Hungarian: possession is shown both by the possessor and by the possessed noun.

Here -ai is part of the possessed plural form.

Breaking it down simply:

• szabály = rule
• szabálya = its rule
• szabályai = its rules

So the ending shows both:

• possession
• plural

You do not need to think of -ai as one neat English word. It is better to learn the full form:

• szabálya = its rule
• szabályai = its rules

Because a társadalom is being used as a normal noun phrase, and Hungarian usually uses the definite article in cases like this.

• a = the
• társadalom = society

Even though English often says just society in a general sense, Hungarian commonly says a társadalom.

So:

• A társadalomnak vannak szabályai. sounds natural for Society has rules.

More or less, yes.

A very literal rendering would be something like:

To society there are its rules.

But that sounds unnatural in English, so the normal translation is:

Society has rules.

Also, Hungarian often uses possessive forms where English prefers a simpler expression. So szabályai does not necessarily sound especially emphatic or poetic; it is just the normal grammar of possession.

Because A társadalom szabályai is only a noun phrase, not a complete sentence.

It means:

• society’s rules
• the rules of society

But it does not mean Society has rules by itself.

To make a full sentence expressing possession, Hungarian normally uses:

A társadalomnak vannak szabályai.

So the difference is:

• A társadalom szabályai = society’s rules
• A társadalomnak vannak szabályai. = society has rules

Yes, but the emphasis changes.

The neutral order here is:

A társadalomnak vannak szabályai.

This presents society as the topic and says something about it.

Other orders are possible, for example:

• Szabályai vannak a társadalomnak.
• A társadalomnak szabályai vannak.

These are more marked and may sound emphatic, contrastive, or stylistically different depending on context.

For a learner, the safest neutral version is:

A társadalomnak vannak szabályai.

Because the sentence is talking about rules in a general, indefinite sense.

So:

• vannak szabályai = it has rules / it has some rules

If you made the noun more definite, the meaning would change.

Hungarian often leaves plural nouns without an article in this kind of existential possession sentence when the meaning is general or indefinite.

Yes, absolutely. It is one of the most important Hungarian sentence patterns.

A useful model is:

X-nek van Y-ja / Y-je
X-nek vannak Y-jai / Y-jei

Examples:

• A fiúnak van kutyája. = The boy has a dog.
• A fiúnak vannak barátai. = The boy has friends.
• A háznak van kertje. = The house has a garden.
• A háznak vannak ablakai. = The house has windows.

Your sentence follows exactly this pattern:

A társadalomnak vannak szabályai.