Competența aceasta este importantă.

Because Romanian usually puts the definite article at the end of the noun.

• competență = a competence / competence
• competența = the competence / the skill

So competența aceasta literally works like this competence/skill, with competența already meaning the competence.

In Romanian, demonstratives like this can often go after the noun:

• competența aceasta = this competence
• cartea aceasta = this book

This is a very common pattern in everyday Romanian.
You can also put the demonstrative before the noun:

• această competență

Both mean this competence, but the postposed version competența aceasta is extremely natural and common.

They usually mean the same thing: this competence.

The main difference is style and structure:

• competența aceasta: noun + demonstrative, very common and natural
• această competență: demonstrative + noun, also correct, sometimes a bit more formal or simply a different way of phrasing it

So:

• Competența aceasta este importantă.
• Această competență este importantă.

Both are correct.

Romanian demonstratives change form depending on position and grammar.

For feminine singular:

• această competență = this competence when the demonstrative comes before the noun
• competența aceasta = this competence when it comes after the noun

So această and aceasta are related forms, but they are used in different structures.

Because adjectives in Romanian agree with the noun in gender and number.

Here, competența is:

• feminine
• singular

So the adjective must also be feminine singular:

• masculine singular: important
• feminine singular: importantă
• masculine plural: importanți
• feminine plural: importante

That is why the sentence uses importantă.

In standard Romanian, este is normally required in a sentence like this.

• Competența aceasta este importantă. = correct
• Competența aceasta importantă. = not standard Romanian

Unlike some languages, Romanian usually keeps the verb to be in the present tense in this kind of sentence.

Yes. E is a shorter, very common form of este.

So these both work:

• Competența aceasta este importantă.
• Competența aceasta e importantă.

The version with este may sound a bit more careful or formal, while e is very common in everyday speech.

These two letters are often new for English speakers:

• ț is pronounced like ts in cats
• ă is a short central vowel, similar to the a in sofa for many English speakers

So:

• competența sounds roughly like kom-pe-TEN-tsa
• aceasta sounds roughly like a-CHAY-a-sta or a-CHES-ta depending on rhythm and accent in real speech
• importantă ends with that short ă sound

The stress is usually:

• compeTENța
• aCEASta
• imporTANtă

It is in the nominative here, because it is the subject of the sentence.

• Competența aceasta = the thing being talked about
• este importantă = what is being said about it

In Romanian, the nominative and accusative often look the same, so learners usually identify the case from the sentence function.

They are closely related, but asta is more informal.

• aceasta = more standard/full form
• asta = very common in speech, more casual

So you may hear:

• Competența asta e importantă.

This is perfectly natural in conversation.
The original sentence with aceasta sounds a bit more neutral or careful.

Because Romanian usually uses an enclitic definite article, which means the article is attached to the end of the noun instead of standing alone before it.

Compare:

• English: the competence
• Romanian: competența

This is one of the big structural differences between Romanian and English. The sentence Competența aceasta este importantă shows that clearly: the noun already carries the meaning of the.