Эта королева очень известна.

Because королева is a feminine singular noun in the nominative case, and эта has to agree with it.

Russian this changes form:

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

So:

• этот король = this king
• эта королева = this queen

Agreement is a big part of Russian grammar: words like this, my, good, etc. often change to match the noun.

It is in the nominative case.

That is because королева is the subject of the sentence: the queen is the person being described as very famous.

A good basic pattern is:

• [subject in nominative] + description

So in Эта королева очень известна, эта королева is the subject.

In the present tense, Russian usually omits the verb to be.

So English This queen is very famous becomes simply:

• Эта королева очень известна.

There is no separate present-tense is in normal Russian sentences like this.

Compare:

• Она умна. = She is smart.
• Он дома. = He is at home.
• Эта королева очень известна. = This queen is very famous.

In past and future tenses, forms of to be do appear:

• Эта королева была очень известна. = This queen was very famous.
• Эта королева будет очень известна. = This queen will be very famous.

Because известна is the short form of the adjective, and short-form adjectives are commonly used when the adjective is the main predicate of the sentence.

Here, the sentence means:

• This queen is very famous

So Russian prefers the short form:

• королева очень известна

The long form известная is more often used before a noun as a regular describing adjective:

• известная королева = a famous queen / the famous queen

A useful contrast:

• Эта королева очень известна. = This queen is very famous.
• Это известная королева. = This is a famous queen.

Because it is a short-form adjective in the feminine singular, matching королева, which is feminine.

Short-form adjectives change by gender and number:

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

Examples:

• Король известен. = The king is famous.
• Королева известна. = The queen is famous.
• Имя известно. = The name is known/famous.
• Эти люди известны. = These people are famous.

So the -а tells you the predicate adjective agrees with a feminine subject.

Очень means very.

It usually goes before the adjective or adverb it modifies:

• очень известна = very famous
• очень красивая = very beautiful
• очень быстро = very quickly

So in this sentence:

• Эта королева очень известна.
• literally: This queen very famous

That is the normal and natural position.

Yes, Russian word order is more flexible than English, but the neutral order here is:

• Эта королева очень известна.

Other orders are possible, but they change the emphasis:

• Очень известна эта королева. = This queen is very famous.
  (more emphasis on very famous)
• Известна эта королева. = This queen is famous.
  (marked, context-dependent)

For a learner, the safest default is the original order:

• Эта королева очень известна.

No. Эта specifically means this.

Russian does not have articles like a and the, so context often tells you whether a noun is definite or indefinite.

Compare:

• Королева очень известна. = The queen is very famous / A queen is very famous
  (depends on context)
• Эта королева очень известна. = This queen is very famous

So эта is not an article. It is a demonstrative word meaning this.

A helpful stress-marked version is:

Э́та короле́ва о́чень известна́.

Approximate pronunciation:

• Эта = EH-ta
• королева = ka-ra-LYE-va
• очень = O-chen
• известна = eez-vees-NA

A rough full approximation:

EH-ta ka-ra-LYE-va O-chen eez-vees-NA

A few pronunciation notes:

• в in королева sounds like v
• ч in очень sounds like ch
• the stress in известна́ is on the last syllable

Usually it works very well as famous in sentences like this, but its broader sense is closer to well-known or known.

So depending on context, известна can mean:

• famous
• well-known
• known

In Эта королева очень известна, the natural English translation is indeed This queen is very famous, but it is useful to remember that the Russian word has a slightly broader range than English famous sometimes does.