Este libro pertenece al vecino.

In Spanish, este is a demonstrative adjective meaning this. When you use este, it already does the job that the would do, so you do not add el before it.

• este libro = this book
• el libro = the book

So el este libro is not correct.

Pertenece comes from the verb pertenecer, which means to belong.

So:

• Este libro pertenece al vecino = This book belongs to the neighbor

You could also say:

• Este libro es del vecino = This book is the neighbor’s / This book belongs to the neighbor

Both are correct, but pertenecer a is a more direct match for to belong to.
Es de is often more common in everyday speech.

Because in Spanish, a + el contracts to al.

So:

• a + el vecino → al vecino

This contraction is mandatory.

Compare:

• pertenece al vecino = belongs to the neighbor
• voy al parque = I’m going to the park

One important exception: if el is part of a proper name, you do not contract it.

• Voy a El Escorial
  not al Escorial

The verb pertenecer is followed by a, just like belong to in English is followed by to.

So the pattern is:

• pertenecer a alguien = to belong to someone

Examples:

• Este coche pertenece a Marta.
• La casa pertenece a mis abuelos.

You have to learn pertenecer a as a set pattern.

Pertenece is the third-person singular present tense form of pertenecer.

Present tense of pertenecer:

• yo pertenezco
• tú perteneces
• él/ella/usted pertenece
• nosotros/nosotras pertenecemos
• vosotros/vosotras pertenecéis
• ellos/ellas/ustedes pertenecen

In your sentence, the subject is este libro, which is singular, so you use pertenece.

Vecino means neighbor and is the masculine singular form.

• el vecino = the male neighbor / the neighbor if masculine
• la vecina = the female neighbor

So the sentence specifically refers to a male neighbor, or to a neighbor identified with masculine grammar.

If it were a female neighbor, it would be:

• Este libro pertenece a la vecina.

El vecino means the neighbor. The definite article is used because the speaker is referring to a specific neighbor, not just any neighbor.

• al vecino = to the neighbor
• a un vecino = to a neighbor

So:

• Este libro pertenece al vecino = This book belongs to the neighbor
• Este libro pertenece a un vecino = This book belongs to a neighbor

The meaning changes from specific to non-specific.

Here, este is a demonstrative adjective because it comes directly before a noun:

• este libro = this book

If it stood alone and replaced the noun, it would function as a pronoun:

• Este pertenece al vecino. = This one belongs to the neighbor.

In modern Spanish spelling, these forms usually do not take accent marks.

In standard Peninsular Spanish, a typical pronunciation would be roughly:

• Este → ES-te
• libro → LEE-bro
• pertenece → per-te-NE-the
• al vecino → al be-THEE-no

A key Spain feature is that c before e or i is often pronounced like the th in think:

• pertenece
• vecino

Also, v and b are pronounced very similarly in Spanish.

Yes, but the neutral, most common order is:

• Este libro pertenece al vecino.

Spanish word order is more flexible than English, but changing it usually changes the emphasis.

For example:

• Al vecino pertenece este libro.
  This sounds more marked or literary, with emphasis on al vecino.

For everyday use, stick with Este libro pertenece al vecino.

Yes. That is a very natural alternative.

• Este libro pertenece al vecino.
• Este libro es del vecino.

Both mean essentially the same thing.
The second uses:

• de + el → del

So del vecino means of the neighbor / the neighbor’s.

In everyday conversation, many speakers would probably use es del vecino more often than pertenece al vecino.

Spanish does not show possession with an apostrophe ’s. Instead, it usually uses de.

So English:

• the neighbor’s book

often becomes Spanish:

• el libro del vecino

Literally, that is closer to the book of the neighbor.

Your sentence uses a different structure:

• Este libro pertenece al vecino = This book belongs to the neighbor

Both are normal ways to express possession in Spanish.