A tanár a könyvtárban olvas, a diák pedig a tanteremben tanul.

A is the Hungarian definite article, meaning the.

Hungarian normally repeats the article before each definite noun phrase, so you get:

• a tanár = the teacher
• a könyvtárban = in the library
• a diák = the student
• a tanteremben = in the classroom

Also, Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Since tanár, könyvtárban, diák, and tanteremben all begin with consonant sounds, a is used each time.

Because Hungarian usually expresses in with a case ending, not with a separate preposition.

So:

• könyvtár = library
• könyvtárban = in the library

and

• tanterem = classroom
• tanteremben = in the classroom

The endings -ban and -ben are called the inessive case, and they mean in.

This is very common in Hungarian: what English often does with prepositions, Hungarian often does with suffixes.

This is because of vowel harmony, a basic feature of Hungarian.

Very roughly:

• words with mostly back vowels usually take -ban
• words with mostly front vowels usually take -ben

Examples here:

• könyvtár has back-vowel pattern overall, so: könyvtárban
• tanterem has front-vowel pattern overall, so: tanteremben

So the suffix changes shape to match the word it attaches to.

Pedig adds a contrast or comparison between the two clauses.

In this sentence, it means something like:

• while
• whereas
• and ... on the other hand
• as for ...

So the sentence has the sense:

• The teacher is reading in the library, while the student is studying in the classroom.

It is not exactly the same as a plain and; it highlights a contrast between the two people and what they are doing.

That is normal Hungarian usage.

Pedig often appears after the first element of the clause, especially after the topic. So:

• a diák pedig ...

is more natural than putting pedig at the very beginning.

You can think of a diák as the topic of the second clause, and pedig comes right after it to mark contrast.

Hungarian word order is flexible, but it is not random. It often reflects topic and focus.

In this sentence, the structure is very natural:

• A tanár = topic
• a könyvtárban = extra information about place
• olvas = the action

and then:

• a diák pedig = topic + contrast marker
• a tanteremben = place
• tanul = action

So the verbs come last here because that is a natural neutral order in this context.

But Hungarian does not always put the verb last. Word order can change if you want to emphasize something different.

Often yes, but the emphasis may change.

For example:

• A tanár a könyvtárban olvas. = neutral, natural
• A könyvtárban a tanár olvas. = emphasizes in the library
• Olvas a tanár a könyvtárban. = possible, but less neutral and more context-dependent

So Hungarian word order is flexible, but different orders can sound more marked or emphasize different parts of the sentence.

Hungarian often leaves out subject pronouns when they are not needed.

Here, the nouns themselves tell you who the subjects are:

• a tanár = the teacher
• a diák = the student

Also, the verb forms show third person singular:

• olvas = he/she reads
• tanul = he/she studies/learns

Since the subject is already stated, Hungarian does not need an extra pronoun.

This is a very useful grammar point.

Hungarian verbs can use two main conjugation patterns:

• indefinite conjugation
• definite conjugation

Here, olvas is the indefinite form because no definite object is stated.

So:

• A tanár a könyvtárban olvas. = The teacher is reading.

But if you add a definite object, you often use the definite form:

• A tanár a könyvet olvassa. = The teacher is reading the book.

In your sentence, there is no object like the book, so olvas is the natural form.

Because the sentence contains two coordinated clauses:

• A tanár a könyvtárban olvas
• a diák pedig a tanteremben tanul

Hungarian normally uses a comma between such clauses, especially when they are joined by a connector like pedig.

So the comma helps separate the two parallel parts of the sentence clearly.