A könyv a polcon van, a kabát pedig a szekrényben.

Because a is the Hungarian definite article, meaning the.

In this sentence:

• a könyv = the book
• a polcon = on the shelf literally on the shelf
• a kabát = the coat
• a szekrényben = in the wardrobe / closet

Hungarian usually repeats the article where English might also use the. The form a is used before words beginning with a consonant. Before a vowel, Hungarian uses az instead.

These are two different location endings:

• -on / -en / -ön = on
• -ban / -ben = in

So:

• polc = shelf
• polcon = on the shelf

and:

• szekrény = wardrobe / closet / cabinet
• szekrényben = in the wardrobe

Hungarian often expresses ideas like on, in, at with noun endings rather than separate prepositions.

That is because of vowel harmony, a major feature of Hungarian.

Very roughly:

• words with back vowels tend to take endings like -ban, -on
• words with front vowels tend to take endings like -ben, -en, -ön

Here:

• polc contains o, a back vowel, so it becomes polcon
• szekrény contains front vowels, so it becomes szekrényben

You do not need to memorize every form separately, but you do need to get used to matching the suffix to the vowels in the word.

Because Hungarian chooses the location ending based on the kind of space involved, much like English does.

• A shelf is treated as a surface, so Hungarian uses -on/-en/-ön
• A wardrobe/closet is treated as an inside space, so Hungarian uses -ban/-ben

So the choice is about meaning:

• polcon = on a surface
• szekrényben = inside something

Pedig is a conjunction that often means something like:

• and
• while
• whereas
• as for
• and as for

In this sentence, it connects the second clause and adds a slight contrast or topic shift:

• The book is on the shelf, and/as for the coat, it is in the wardrobe.

It is not exactly the same as és (and).
Pedig often sounds a little more contrastive or comparative.

Yes, you could say:

A könyv a polcon van, és a kabát a szekrényben van.

That would still be correct. But the nuance is a bit different:

• és = plain and
• pedig = and/while/as for, often with contrast or comparison

So pedig is very natural here because the sentence is comparing the locations of two different things.

The full version could be:

A könyv a polcon van, a kabát pedig a szekrényben van.

In the original sentence, the second van is omitted because it is already understood. This is a very normal kind of ellipsis in Hungarian, especially when the same verb would just be repeated.

So both are possible:

• ... a szekrényben. = more concise
• ... a szekrényben van. = fuller, also correct

That is a very common and very good question.

In Hungarian, the present-tense third-person form of to be is often omitted with noun and adjective predicates:

• A könyv érdekes. = The book is interesting.
• Ő tanár. = He/She is a teacher.

But with location, Hungarian normally uses van:

• A könyv a polcon van. = The book is on the shelf.

So:

• adjective/noun predicate → often no van
• location → usually van

This is a normal neutral Hungarian word order for a sentence about location:

• A könyv = topic, what we are talking about
• a polcon = where it is
• van = is

So the pattern is roughly:

[topic] + [location] + van

Hungarian word order is flexible, but different orders create different emphasis.

For example:

• A könyv a polcon van. = neutral
• A polcon van a könyv. = stronger emphasis on where the book is

The sentence you were given is the natural neutral version.

Yes, very important.

In Hungarian, accent marks are not optional spelling decorations. They show real differences in:

• vowel quality
• often vowel length

For example:

• a and á are different vowels
• o and ö are different vowels
• e and é are different vowels

So you should learn and write them carefully:

• könyv, not konyv
• kabát, not kabat
• szekrény, not szekreny

Leaving them out can make the word incorrect or harder to understand.

Yes, if you follow the Hungarian word order very closely, it looks like that.

Hungarian does not arrange the sentence the same way English does. A very literal breakdown is:

• A könyv = the book
• a polcon = on the shelf
• van = is

So literally it is something like:

The book on the shelf is.

But in normal English, we say:

The book is on the shelf.

This is a good example of why it is better to understand Hungarian sentence structure directly, instead of translating word by word.