A könyve még mindig a polcon van.

Könyv means book. Könyve is the possessed form, meaning his/her/its book.

Hungarian often shows possession by adding a suffix to the noun:

• könyvem = my book
• könyved = your book
• könyve = his/her book

So the -e ending is what makes könyv become könyve.

Because Hungarian can use the definite article with possessed nouns.

So a könyve is normal Hungarian. English usually says just his book or her book, without the, but Hungarian often says a könyve when the noun phrase is definite and specific.

In other words:

• könyve = his/her book
• a könyve = the specific book of his/hers, or simply his/her book in normal context

The form könyve only tells you that the possessor is 3rd person singular. It does not show gender.

So it can mean:

• his book
• her book
• its book

You find out which one is meant from the context.

Még mindig is a fixed expression meaning still.

The two words separately have broader meanings:

• még can mean still, yet, or more
• mindig means always

But together, még mindig specifically means that something continues to be true: still.

So here it means the book has remained on the shelf.

Hungarian often uses endings instead of prepositions.

• polc = shelf
• polcon = on the shelf

The ending -on here expresses location on something. This is called the superessive case.

Compare:

• a polcon = on the shelf
• a polcra = onto the shelf
• a polcról = off the shelf / from the shelf

So a polcon van means is on the shelf.

Because Hungarian keeps the article, and the case ending goes on the noun.

So:

• a polc = the shelf
• a polcon = on the shelf

The article a stays the same, and the location ending is added to polc.

Hungarian often drops van in the present tense, 3rd person, but not in every kind of sentence.

It is usually omitted when the predicate is a noun or adjective:

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

But with location, van is normally kept:

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

So in this sentence, van is needed.

Hungarian word order is more flexible than English word order.

This sentence has a natural, neutral structure:

• A könyve = the topic
• még mindig = still
• a polcon = the location
• van = is

Putting van at the end is very normal in sentences like this. Other word orders are possible, but they would change the emphasis rather than the basic meaning.