Эта компетенция важна для работы.

Because компетенция is a feminine noun in Russian, and the demonstrative this has to agree with it.

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

So:

• эта компетенция = this competency / this skill area

A good clue is that компетенция ends in -я, which is a very common feminine ending.

It is similar, but not always identical in usage.

Компетенция can mean:

• an area of competence
• a skill set
• expertise
• responsibility or sphere of authority, depending on context

In this sentence, it most naturally means something like:

• this competency
• this skill
• this area of expertise

So a learner should be careful: компетенция does not always map perfectly onto the English word competence in every context.

Because Russian is using the short form adjective here.

There are two common adjective forms in Russian:

1. long form: важный, важная, важное
2. short form: важен, важна, важно, важны

In sentences like X is important, Russian very often uses the short form:

• Он важен. = He is important.
• Она важна. = She is important.
• Эта компетенция важна. = This competency is important.

The short form is especially common when the adjective is being used as the main predicate of the sentence.

Because it agrees with компетенция, which is feminine.

Russian adjectives must match the noun in gender and number. Since компетенция is feminine singular, the short-form adjective is also feminine singular:

• masculine: важен
• feminine: важна
• neuter: важно
• plural: важны

So:

• компетенция важна
• not компетенция важен

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

So where English says:

• This competency is important

Russian simply says:

• Эта компетенция важна

There is no present-tense is written or spoken in normal Russian sentences like this.

In other tenses, Russian does use forms of to be:

• Эта компетенция была важна = This competency was important
• Эта компетенция будет важна = This competency will be important

Для requires the genitive case, so работа changes to работы.

• dictionary form: работа
• after для: для работы

So:

• для работы = for work

This is a very common pattern in Russian:

• для дома = for the house / for home
• для студента = for the student
• для книги = for the book

It can often be translated in several similar ways depending on context:

• for work
• for the job
• for working
• for one’s work

In this sentence, для работы most naturally means something like:

• important for work
• important in doing the job
• important for professional activity

Russian often leaves this a little broader than English.

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

• Эта компетенция важна для работы.

You could also hear:

• Для работы эта компетенция важна.

This version puts more emphasis on for work.

Even though the order can change, the endings still show the grammar, so the meaning remains basically the same.

Yes, grammatically you can, but it sounds more formal, heavier, and less natural in ordinary speech.

• Эта компетенция важна для работы. = simple, natural
• Эта компетенция является важной для работы. = more formal, bureaucratic, or academic

Also note that with является, Russian usually uses the instrumental case on the adjective:

• важной

So for normal conversation and most writing, the shorter version is better.

Both words exist, but they are not exactly the same.

• компетенция often refers to a specific competency, area of expertise, or responsibility
• компетентность usually means competence as a general quality or level of ability

So:

• Эта компетенция важна для работы. = This competency is important for the job.
• Компетентность важна для работы. = Competence is important for work.

The first points to one particular skill or area; the second talks about competence in a broader, more abstract way.

Depending on context, a natural English translation could be:

• This competency is important for the job.
• This skill is important for work.
• This area of expertise is important for the work.

Even if the meaning has already been given, it is useful to know that English may prefer skill over competency in many everyday contexts, while Russian компетенция is still perfectly normal here.