Anna hat eine fantastische Idee.

Hat is the 3rd person singular present-tense form of the verb haben (to have).

Because the subject is Anna = she, German uses:

• ich habe = I have
• du hast = you have
• er/sie/es hat = he/she/it has

So Anna hat means Anna has.

In a normal German main clause, the finite verb usually comes in position 2. This is often called the V2 rule.

In this sentence:

• Anna = position 1
• hat = position 2
• eine fantastische Idee = the rest of the sentence

So the pattern is:

Anna | hat | eine fantastische Idee

If you move another element to the front, the verb still stays second:

• Heute hat Anna eine fantastische Idee.
• Eine fantastische Idee hat Anna.

Because Idee is a feminine noun.

The singular indefinite articles are:

• ein for masculine and neuter
• eine for feminine

So:

• ein Tisch = a table, masculine
• ein Buch = a book, neuter
• eine Idee = an idea, feminine

You simply have to learn the noun together with its gender:
die Idee = the idea

It is the accusative case, because it is the direct object of hat.

In Anna hat eine fantastische Idee, Anna is the subject, and the thing she has is the object.

So:

• Anna = subject, nominative
• eine fantastische Idee = direct object, accusative

A useful thing to know: for feminine singular, the indefinite article is eine in both nominative and accusative, so you do not see a change here.

Compare that with masculine:

• ein Mann = a man, nominative
• einen Mann = a man, accusative

Because adjectives placed before a noun in German usually need an ending.

The basic adjective is fantastisch.
Before Idee, it becomes fantastische.

That happens because the adjective has to agree with the noun phrase. Here the noun phrase is:

• eine = feminine singular article
• Idee = feminine singular noun
• case = accusative

After eine, the correct adjective ending here is -e:

• eine fantastische Idee

This is a very common pattern in German adjective endings.

Because all nouns are capitalized in German.

So in this sentence:

• Anna is capitalized because it is a proper name
• Idee is capitalized because it is a noun

This is one of the most noticeable spelling differences between German and English.

A few useful pronunciation points:

• ei is usually pronounced like English eye
• ie is usually pronounced like English ee
• a final -e is often a weak uh sound

So roughly:

• eine ≈ EYE-nuh
• Idee ≈ ee-DAY

Also notice the difference:

• ei = eye
• ie = ee

That contrast is very important in German.

The sentence you have is the most neutral, standard order:

Anna hat eine fantastische Idee.

But German word order is somewhat flexible for emphasis. For example:

• Eine fantastische Idee hat Anna.
  This puts extra focus on eine fantastische Idee.

• Heute hat Anna eine fantastische Idee.
  Here Heute comes first, but hat still stays in second position.

So yes, the order can change, but in a normal simple statement, subject + verb + rest is the most natural choice.