Этот принц очень известен.

Because этот has to match принц in gender, number, and case.

• принц is masculine
• singular
• nominative case (the subject of the sentence)

So the correct form is этот.

Related forms:

• эта = feminine singular
• это = neuter singular
• эти = plural

For example:

• эта принцесса = this princess
• это письмо = this letter
• эти люди = these people

Принц is in the nominative singular because it is the subject of the sentence: the person being talked about.

In Russian, the subject of a basic sentence is usually in the nominative case.

So:

• Этот принц = the subject
• очень известен = what is being said about him

If prince had a different role in the sentence, its form could change. For example:

• Я вижу этого принца. = I see this prince.
  Here принца is accusative.

In the present tense, Russian normally leaves out the verb to be.

So English:

• This prince is very famous

becomes Russian:

• Этот принц очень известен.

There is no separate word for is here.

This is completely normal in Russian. Compare:

• Он врач. = He is a doctor.
• Она счастлива. = She is happy.

But in the past or future, forms of to be do appear:

• Этот принц был очень известен. = This prince was very famous.
• Этот принц будет очень известен. = This prince will be very famous.

This is one of the most common learner questions.

Известен is the short form of the adjective известный.

Russian adjectives can have:

• a full form: известный
• a short form: известен

In this sentence, the adjective is used as a predicate: it tells us what the prince is like. In that role, Russian often uses the short form, especially in more standard or neutral written language.

So:

• Этот принц очень известен. = This prince is very famous.

The full form известный is more commonly used before a noun:

• известный принц = a famous prince

You may sometimes hear full forms used after the noun too, but in a sentence like this, известен is the more standard choice.

Очень means very.

It goes before the adjective it modifies:

• очень известен = very famous

This is the normal position in Russian, just like in English:

• very famous
• очень известен

You can use очень with many adjectives and adverbs:

• очень красивый = very beautiful
• очень хорошо = very well
• очень интересный = very interesting

Yes. Russian word order is more flexible than English word order.

The neutral, straightforward order is:

• Этот принц очень известен.

But other orders are possible for emphasis:

• Очень известен этот принц.
• Этот принц известен очень. — possible in special contexts, but not the normal choice
• Известен этот принц очень. — unusual and strongly marked

Even though the order can change, the original sentence sounds the most natural in ordinary speech and writing.

So for a learner, the best default is:

• subject + adverb + predicate adjective
• Этот принц очень известен.

The short-form adjective changes to match the noun’s gender and number.

Since принц is masculine singular, the adjective is:

• известен

Other forms are:

• masculine: известен
• feminine: известна
• neuter: известно
• plural: известны

Examples:

• Этот принц известен. = This prince is famous.
• Эта принцесса известна. = This princess is famous.
• Это имя известно. = This name is famous / well known.
• Эти люди известны. = These people are famous.

It can mean either, depending on context.

The adjective известный / известен often covers both:

• famous
• well-known
• known

So:

• Этот принц очень известен.

could be understood as:

• This prince is very famous.
• This prince is very well-known.

The exact English translation depends on context, but the Russian grammar stays the same.

A helpful pronunciation guide is:

Э́тот принц о́чень изве́стен.

Approximate pronunciation:

• этот ≈ EH-tuht
• принц ≈ prints
• очень ≈ OH-chin or OH-chyen (depending on how precisely you describe it)
• известен ≈ iz-VYE-stin

Stress:

• Э́тот
• о́чень
• изве́стен

A few tips:

• ц in принц sounds like ts
• unstressed о is often reduced in pronunciation
• the final consonants in Russian are often less strongly released than in English

Yes, but it means something different.

• Этот принц очень известен. = This prince is very famous.
  This is a complete sentence.

• Этот известный принц = this famous prince
  This is just a noun phrase, not a full sentence by itself.

Here известный is the full adjective placed before the noun, where full adjective forms are normally used.

So:

• известный принц = a famous prince
• принц известен = the prince is famous

You would change both the demonstrative and the short-form adjective so they agree with the noun.

For принцесса:

• Эта принцесса очень известна. = This princess is very famous.

Changes:

• этот → эта because принцесса is feminine
• известен → известна because the adjective must also be feminine

This kind of agreement is a central feature of Russian grammar.