Эта книга принадлежит другу.

Because книга is a feminine singular noun, and the demonstrative this has to agree with it.

• masculine: этот
• feminine: эта
• neuter: это

So:

• эта книга = this book

If the noun were masculine, you would use этот, for example этот стол (this table).

Because the verb принадлежать (to belong) requires the dative case for the person or thing that something belongs to.

So in this sentence:

• книга is the thing doing the grammatical action
• другу is the person to whom the book belongs

That is why друг changes to другу.

A useful pattern is:

• X принадлежит Y = X belongs to Y

where Y is usually in the dative.

Книга is in the nominative case because it is the grammatical subject of the sentence.

In Эта книга принадлежит другу:

• эта книга = subject, so nominative
• другу = dative, because of принадлежит

Принадлежит is the 3rd person singular form of the verb принадлежать (to belong).

It matches книга, which is singular:

• я принадлежу
• ты принадлежишь
• он / она принадлежит
• мы принадлежим
• вы принадлежите
• они принадлежат

Since книга is a singular noun, you use принадлежит.

Russian has no articles like English a / an / the.

So Эта книга принадлежит другу can be understood as something like:

• This book belongs to a friend
• This book belongs to the friend
• This book belongs to my/your/etc. friend

The exact meaning depends on context. In this sentence, эта already makes книга specific: this book.

By itself, другу just means to a friend / to the friend / to [someone’s] friend, depending on context.

Russian often leaves this kind of information unstated unless it is important. If you want to be more specific, you can add more words:

• моему другу = to my friend
• этому другу = to this friend
• одному другу = to a friend

So the bare другу is natural, but somewhat context-dependent in English.

Yes. Russian word order is more flexible than English because case endings show the grammatical roles.

The neutral, straightforward order is:

• Эта книга принадлежит другу.

But you may also see:

• Другу принадлежит эта книга.
• Эта книга другу принадлежит.

These alternatives can shift emphasis or sound more literary or contrastive. For learners, the original order is the safest and most natural default.

Yes, very often Russian prefers a construction with у:

• Эта книга у друга.

Literally, this is more like This book is with the friend, but in many contexts it can mean The friend has this book or This book is in the friend’s possession.

The difference is roughly:

• принадлежит = emphasizes ownership
• у construction = often emphasizes possession / where the item is / who has it

So Эта книга принадлежит другу sounds specifically like This book belongs to a friend.

A practical pronunciation guide is:

• pri-nad-lee-ZHEET

A few notes:

• the stress is on the last syllable: -жит
• ж sounds like the s in measure
• unstressed vowels are reduced in natural speech

If you want to say the whole sentence approximately:

• EH-ta KNEE-ga pree-nad-lee-ZHEET DROO-goo

This is only an approximation, but it can help at first.

Russian often expresses possession differently from English.

English commonly says:

• This book is the friend’s
• This is a friend’s book

Russian often uses:

• Эта книга принадлежит другу
• literally: This book belongs to a friend

Russian can also use possessive-style expressions, but the принадлежать + dative pattern is very normal when talking about ownership.

Yes, but it is somewhat more formal and explicit than some other ways of expressing possession.

You will hear it in statements about ownership, categories, or affiliation, for example:

• Этот дом принадлежит моей семье. = This house belongs to my family.
• Книга принадлежит библиотеке. = The book belongs to the library.

In everyday conversation, Russians also very often use:

• у меня есть...
• это моё
• это у друга

So принадлежит is common and useful, but not the only way to talk about possession.