Mindkettő gyertyatartó: az egyik hibátlan, a másikon karc van.

Mindkettő means both (of them) and can stand on its own as a pronoun.

• Mindkettő gyertyatartó. = Both are candlesticks.

By contrast, mindkét goes directly before a noun:

• mindkét gyertyatartó = both candlesticks

So a useful rule is:

• mindkettő = standalone both
• mindkét + noun = both + noun

In Hungarian, the present-tense 3rd person forms of to be are usually omitted in sentences like X is Y or X is adjective.

So:

• Mindkettő gyertyatartó. literally looks like Both candlestick
• but it means Both are candlesticks

This is normal Hungarian grammar.

Compare:

• A ház nagy. = The house is big.
• Ő tanár. = He/She is a teacher.

But van does appear when it really means there is / exists / is located, which is why the second clause has van: a másikon karc van.

Because gyertyatartó is a predicate noun here: it tells you what the subject is.

In Hungarian, predicate nouns used for classification usually do not take an article:

• Ő orvos. = He/She is a doctor.
• Ez könyv. would be unnatural, but in a fuller sentence like Ez egy könyv the egy can appear when introducing something as a book.
• Mindkettő gyertyatartó. = Both are candlesticks.

If you used a gyertyatartó, it would sound more like the candlestick, referring to a specific one, not just naming the type of object.

Hungarian often keeps this kind of predicate noun in the singular when the idea is each one is X.

So Mindkettő gyertyatartó is understood as:

• each of the two is a candlestick
• therefore both are candlesticks

This singular form is very natural here.

Yes. It is made from:

• gyertya = candle
• tartó = holder

So gyertyatartó literally means candle-holder, i.e. candlestick or candleholder.

Hungarian forms a lot of everyday nouns this way.

This is the normal paired pattern for one ... the other ... in Hungarian.

• az egyik = one of the two / one
• a másik = the other

In your sentence, másik has a case ending, so it becomes másikon.

The articles are also normal here:

• az egyik because egyik begins with a vowel sound
• a másikon because másikon begins with a consonant sound

So the structure is very idiomatic:

• az egyik ..., a másik ... = one ..., the other ...

Hibátlan means flawless, perfect, or without defects.

It comes from:

• hiba = mistake, fault, defect
• -tlan / -tlen = a suffix meaning without, lacking

So hibátlan literally means something like without fault.

This suffix is very common in Hungarian, for example:

• hiba → hibátlan = faultless
• íz → íztelen = tasteless
• remény → reménytelen = hopeless

Másikon is másik plus the superessive ending -on / -en / -ön, which usually means on.

So:

• másik = the other one
• másikon = on the other one

In this sentence, that means the scratch is physically on the surface of the other candlestick.

So the clause is literally close to:

• On the other one, there is a scratch.

Because Hungarian often expresses this kind of idea with an existence/location structure rather than a direct have structure.

So instead of saying:

• the other has a scratch

Hungarian says something like:

• on the other one there is a scratch

That is exactly what a másikon karc van does.

This is especially natural when the defect is something physically located on the surface of an object.

Because Hungarian often leaves out egy when it simply introduces something indefinite and the exact number is not important.

So:

• karc van = there is a scratch / it has a scratch
• egy karc van = there is one scratch

Adding egy makes the idea of one more explicit. Without it, the sentence sounds a bit more natural and matter-of-fact: it is simply noting that there is a scratch.

Hungarian word order is strongly shaped by topic and focus, not just by a fixed English-style pattern.

Here:

• a másikon sets the topic or frame: as for the other one / on the other one
• karc van gives the new information: there is a scratch

So the sentence naturally moves from what we are talking about to what is being said about it.

That is why this order feels natural:

• a másikon karc van

Other word orders are possible in Hungarian, but they would change the emphasis.

The colon introduces a more specific breakdown of the general statement.

First the sentence gives the overall point:

• Mindkettő gyertyatartó = Both are candlesticks

Then it explains the difference between them:

• az egyik hibátlan, a másikon karc van = one is flawless, the other has a scratch

So the colon works much like it does in English: it means here is the detail/explanation.