A nyelvi hiba a ruhacímkén van.

It turns nyelv = language into an adjective meaning language-related or linguistic.

So:

• nyelv = language
• nyelvi = linguistic, language-related

That is why Hungarian says nyelvi hiba rather than nyelv hiba.

This -i adjective-forming ending is very common in Hungarian:

• iskola → iskolai = school-related
• Budapest → budapesti = from Budapest, Budapest-related

Not exactly.

• nyelvi hiba is broader. It can mean a language mistake, wording problem, or a general error in the text/language.
• nyelvtani hiba is more specific: a grammar mistake.

So in this sentence, nyelvi hiba does not necessarily mean only grammar. It could be any kind of language-related error on the label.

Because Hungarian very often forms compounds as one word.

Here:

• ruha = clothing, garment
• címke = label, tag

So:

• ruhacímke = clothing label, garment label

English often uses separate words where Hungarian prefers a compound.

It is made like this:

• ruhacímke = clothing label
• ruhacímkén = on the clothing label

The ending here is the superessive case, which expresses on a surface.

With this noun, the final -e becomes -é before the ending:

• címke → címkén

So a ruhacímkén literally works like the clothing-label-on.

Because the article and the case ending do different jobs.

• a = the
• -n / -on / -en / -ön = on

Hungarian keeps the definite article even when the noun has a case ending:

• a házban = in the house
• az asztalon = on the table
• a ruhacímkén = on the clothing label

So a ruhacímkén is completely normal.

Hungarian has two forms of the definite article:

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

Here both following words begin with consonant sounds:

• nyelvi
• ruhacímkén

So a is correct in both places.

For comparison:

• az autón = on the car
• az étlapon = on the menu

That is true in many sentences, but not in this kind.

Hungarian usually omits van in present-tense sentences where something is described by a noun or adjective:

• A címke új. = The label is new.
• A hiba komoly. = The mistake is serious.

But when the sentence says where something is, van is normally kept:

• A hiba a címkén van. = The mistake is on the label.

So in this sentence, van is needed because the sentence expresses location.

Because this is a natural Hungarian word order for a sentence where the location comes right before van.

The structure is roughly:

• A nyelvi hiba = topic, what we are talking about
• a ruhacímkén van = what we say about it

So the sentence feels like:

• As for the language error, it is on the clothing label.

Hungarian word order is flexible, though, and changing it changes the emphasis.

For example:

• A nyelvi hiba a ruhacímkén van. = neutral
• A ruhacímkén van a nyelvi hiba. = stronger emphasis on the location
• Nyelvi hiba van a ruhacímkén. = there is a language error on the clothing label

ruhacímkén is in the superessive case, which usually means on.

Use it for something being on a surface:

• az asztalon = on the table
• a falon = on the wall
• a címkén = on the label

By contrast, -ban/-ben means in:

• a dobozban = in the box
• a táskában = in the bag

A label is treated as a surface, so Hungarian uses -n / -on / -en / -ön, not -ban/-ben.