A blúz a szoknya mellett van.

Because a is the Hungarian definite article, meaning the.

So:

• a blúz = the blouse
• a szoknya = the skirt

Hungarian uses the article separately before each noun, just like English does here.

Also, Hungarian has two forms of the:

• a before a consonant sound
• az before a vowel sound

Since blúz and szoknya both begin with consonant sounds, the article is a in both places.

Mellett means next to, beside, or by in a physical-location sense.

So a szoknya mellett means:

• next to the skirt
• beside the skirt

In this sentence, it tells you where the blouse is located.

No. Mellett is a postposition, not a case ending.

That means it works like a preposition in English, but it comes after the noun instead of before it.

Compare:

• English: next to the skirt
• Hungarian: a szoknya mellett
  literally: the skirt next to/beside

So instead of adding an ending directly to szoknya, Hungarian uses the separate word mellett after it.

Because mellett takes the noun in its basic form here.

So:

• szoknya = skirt
• a szoknya mellett = next to the skirt

Not every Hungarian location expression uses a noun ending. Some use case suffixes, but others use postpositions like mellett, alatt, fölött, előtt, and so on.

That is a very common question. Hungarian often drops van in the present tense, but not in all situations.

With noun or adjective predicates in the 3rd person singular, van is usually omitted:

• A blúz szép. = The blouse is الجميل/beautiful.
• A blúz új. = The blouse is new.

But with location, van is normally kept:

• A blúz a szoknya mellett van. = The blouse is next to the skirt.

So here van is needed because the sentence is saying where something is.

In a neutral Hungarian sentence about location, it is very natural for van to come at the end.

Structure here:

• A blúz = topic / what we are talking about
• a szoknya mellett = location
• van = is

So the sentence flows naturally as:

The blouse — next to the skirt — is.

That sounds strange in English, but it is normal in Hungarian word order.

Yes. Hungarian word order is flexible, but changing the order often changes the focus or emphasis.

For example:

• A blúz a szoknya mellett van.
  Neutral: The blouse is next to the skirt.

• A szoknya mellett van a blúz.
  More like: It is next to the skirt that the blouse is / The blouse is next to the skirt with more focus on the location.

So the original sentence is a straightforward, neutral way to say it.

It can often be translated as:

• The blouse is next to the skirt
• The blouse is beside the skirt
• sometimes The blouse is by the skirt

The most direct and common translation is next to or beside.
By is possible in English, but it can sound a little less precise.

The thing being located is a blúz.

That is the subject of the sentence:

• A blúz = the blouse

The location is:

• a szoknya mellett = next to the skirt

So the sentence tells us the position of the blouse relative to the skirt.

No, blúz is a loanword, ultimately related to blouse.

That is why it may look familiar to an English speaker. Hungarian has many everyday words borrowed from other languages, especially for clothing, culture, and modern life.

Then Hungarian usually uses a suffixed form:

• mellette = next to it

So:

• A blúz mellette van. = The blouse is next to it.

And similarly:

• mellettem = next to me
• melletted = next to you
• mellette = next to him/her/it
• mellettünk = next to us

This is very common with Hungarian postpositions.