Эта принцесса очень известна.

Because принцесса is a feminine singular noun, and the demonstrative этот must agree with it in gender, number, and case.

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

So:

• эта принцесса = this princess

Both words are in the nominative singular feminine form here.

In Russian, nouns have grammatical gender. Принцесса is feminine both in meaning and in grammar.

A big clue is the ending -а, which is very often feminine in Russian.

Because принцесса is feminine, words that describe it or point to it also take feminine forms, such as:

• эта
• известна

In Russian, the verb to be is usually omitted in the present tense.

So English:

• This princess is very famous.

becomes Russian:

• Эта принцесса очень известна.

In the past or future, forms of to be do appear indirectly through other structures:

• Эта принцесса была очень известна. = This princess was very famous.
• Эта принцесса будет очень известна. = This princess will be very famous.

So the missing is is completely normal.

Известна is the short-form adjective, feminine singular, used predicatively here, meaning something like is famous.

It agrees with принцесса, so it must be feminine singular.

Compare:

• известный = masculine long form
• известная = feminine long form
• известна = feminine short form

In this sentence, известна works like the predicate:

• Принцесса известна. = The princess is famous.

This is a very important pattern in Russian.

A very useful distinction:

• известная принцесса = a famous princess / the famous princess

  • here the adjective directly modifies the noun
  • this is the long form

• принцесса известна = the princess is famous

  • here the adjective is part of the predicate
  • this is the short form

So:

• Эта известная принцесса = this famous princess
• Эта принцесса очень известна = this princess is very famous

In modern Russian, long forms can sometimes also appear after the noun in similar meanings, but the short form in sentences like this is very standard and important to recognize.

It is in the nominative case.

Why?

• принцесса is the subject of the sentence
• эта agrees with принцесса
• the sentence is simply stating something about the subject

So both are nominative singular feminine:

• эта
• принцесса

Очень means very.

It intensifies the adjective:

• известна = famous
• очень известна = very famous

It does not change form. It is an adverb, so it stays the same.

Other examples:

• очень красива = very beautiful
• очень умна = very clever

Yes. Russian word order is more flexible than English word order, although the most neutral version here is:

• Эта принцесса очень известна.

Other orders are possible for emphasis, contrast, or style, for example:

• Очень известна эта принцесса.
• Эта принцесса известна.
• Известна эта принцесса.

But these do not all sound equally neutral. For a learner, the original sentence is the safest and most natural default.

No. Russian does not use articles.

So Russian does not have separate words corresponding exactly to a and the.

Instead, Russian shows meaning through context, word order, and words like этот / эта / это / эти when needed.

So:

• принцесса could mean a princess or the princess, depending on context
• эта принцесса clearly means this princess

A helpful pronunciation guide is:

• Эта = EH-ta
• принцесса = prin-TSE-sa
• очень = O-chen
• известна = iz-VYES-na

Approximate full pronunciation:

EH-ta prin-TSE-sa O-chen iz-VYES-na

The main stresses are:

• Э́та
• принце́сса
• о́чень
• изве́стна

That is simply the standard spelling of the word:

• принцесса

It comes from a borrowed word, and Russian keeps the double с in spelling. When writing and reading, just learn it as part of the word.

It is similar to how English keeps doubled consonants in some borrowed or inherited words.