A gyerek sapkája a széken van.

Because Hungarian uses the definite article a / az much like English the.

So in this sentence:

• A gyerek = the child
• a széken = on the chair

Hungarian often repeats the article where English does too, and sometimes even where English might phrase things differently.

A quick note:

• a is used before words beginning with a consonant
• az is used before words beginning with a vowel

So you get:

• a gyerek
• a szék but
• az asztal

Sapka means hat or cap.

Sapkája means his/her hat or the child's hat.

Hungarian usually shows possession by adding a possessive ending to the possessed noun, rather than using a separate word like English of or an apostrophe 's.

Here:

• sapka = hat
• sapkája = his/her hat

In the full phrase:

• a gyerek sapkája = the child's hat

So the possession is shown on sapkája, not on gyerek.

Both patterns exist in Hungarian, but they are used a little differently.

In this sentence:

• a gyerek sapkája is the normal, simple way to say the child's hat

Another possible version:

• a gyereknek a sapkája

This is also grammatical, but it puts more emphasis on the possessor, almost like:

• the hat of the child
• or the child's hat with slightly more focus on whose hat

So:

• a gyerek sapkája = neutral, common
• a gyereknek a sapkája = more explicit/emphatic

That -ja / -je ending is a common third-person singular possessive ending in Hungarian.

So the pattern is often:

• sapka = hat
• sapkája = his/her hat

The -j- is part of the possessive form for many nouns. The exact possessive ending depends on vowel harmony and on the shape of the noun.

Some examples:

• könyv = book → könyve = his/her book
• kabát = coat → kabátja = his/her coat
• sapka = hat → sapkája = his/her hat

Also notice the vowel length change:

• a → á in sapkája

That kind of change is normal in some possessed forms.

Széken means on the chair.

It comes from:

• szék = chair
• széken = on the chair

The ending -on / -en / -ön is a location ending often called the superessive case. It usually means on.

Examples:

• asztalon = on the table
• széken = on the chair
• földön = on the ground

So in the sentence:

• a széken van = is on the chair

Hungarian often uses case endings where English uses prepositions.

Because the -on / -en / -ön ending changes shape depending on the word, following Hungarian vowel harmony and established noun patterns.

For szék, the correct form is:

• szék → széken

You simply have to learn which version a noun takes, although vowel harmony helps a lot.

Compare:

• asztal → asztalon
• szék → széken
• föld → földön

So the ending has three common forms:

• -on
• -en
• -ön

And szék takes -en.

That's a very common question.

Hungarian often leaves out van in the present tense when describing something with an adjective or noun, especially in the third person:

• A gyerek kicsi. = The child is small.
• Ez egy szék. = This is a chair.

But with location, Hungarian normally uses van:

• A gyerek sapkája a széken van. = The child's hat is on the chair.
• A könyv az asztalon van. = The book is on the table.

So:

• descriptions/identity in 3rd person present: often no van
• location/existence: usually van

Yes, Hungarian word order is flexible, but different orders change the emphasis or focus.

The version you have is a very neutral sentence:

• A gyerek sapkája a széken van.
• The child's hat is on the chair.

This presents the child's hat as the topic, and on the chair as the important information about it.

Other word orders are possible, for example:

• A széken van a gyerek sapkája.
  • More like: It is on the chair that the child's hat is.
  • Slightly more focus on the location.

So the original sentence is a natural default way to say it.

Gyerek is singular here, and it means child or kid depending on style.

So:

• a gyerek = the child / the kid

The plural would be:

• a gyerekek = the children / the kids

In this sentence, because the possessed noun is singular too:

• a gyerek sapkája = the child's hat

If it were plural, it would look different.

By context, or by the possessor noun in front of it.

On its own:

• sapkája = his/her hat

But in this sentence:

• a gyerek sapkája

the possessor is explicitly stated: the child

So the phrase means:

• the child's hat

Hungarian does not need a separate word like his or her here, because the possessed form plus the possessor noun already makes the relationship clear.

Yes, it is a very natural and standard sentence.

It sounds like ordinary everyday Hungarian and is a good model for these patterns:

• definite article: a
• possession: gyerek sapkája
• location with case ending: széken
• existential/location verb: van

So it is a very useful sentence for learning several important Hungarian structures at once.