Αυτή η υπόθεση είναι σημαντική.

Yes, together they work like English this.

In Modern Greek, it is very common to use:

• αυτή = the demonstrative (this)
• η = the definite article (the)

So αυτή η υπόθεση literally looks like this the case/assumption, but in natural English it is just this case/assumption.

This pattern is normal Greek:

• αυτό το βιβλίο = this book
• αυτή η μέρα = this day
• αυτές οι ιδέες = these ideas

Because υπόθεση is a feminine singular noun, and the demonstrative must agree with it.

Greek demonstratives change for gender:

• αυτός = masculine
• αυτή = feminine
• αυτό = neuter

Since υπόθεση is feminine, Greek uses αυτή.

You usually learn the gender as part of the noun.

Here, one clue is the article:

• η υπόθεση

The article η marks a feminine singular nominative noun.

So when you see η υπόθεση, you know the noun is feminine.

It is in the nominative singular.

Why? Because it is the subject of the sentence:

• Αυτή η υπόθεση = the thing being talked about

In sentences with είναι (is), the subject is normally in the nominative.

Because adjectives in Greek agree with the noun they describe in:

• gender
• number
• usually case as well

Since υπόθεση is feminine singular, the adjective appears in the feminine singular form:

• masculine: σημαντικός
• feminine: σημαντική
• neuter: σημαντικό

So:

• η υπόθεση είναι σημαντική not
• η υπόθεση είναι σημαντικός

Here it is a predicate adjective.

That means it comes after είναι and describes the subject:

• Αυτή η υπόθεση είναι σημαντική = This case/assumption is important

Compare that with an adjective directly before the noun:

• η σημαντική υπόθεση = the important case/assumption

Both use the adjective σημαντική, but the structure is different.

Είναι is the 3rd person singular present of είμαι = to be.

So here it means is.

A few forms of είμαι:

• είμαι = I am
• είσαι = you are
• είναι = he/she/it is

In this sentence, the subject is Αυτή η υπόθεση, so Greek uses είναι.

In a normal standard sentence like this, no—you should keep είναι.

So:

• Αυτή η υπόθεση είναι σημαντική = correct

Leaving out είναι would sound incomplete in standard Modern Greek.

Greek often drops subject pronouns because the verb ending shows the person, but it does not normally drop the verb to be in this kind of sentence.

The neutral, most natural order is:

• Αυτή η υπόθεση είναι σημαντική.

But Greek word order is more flexible than English, especially for emphasis.

For example:

• Η υπόθεση αυτή είναι σημαντική.
  Also correct; slightly different style/emphasis.
• Σημαντική είναι αυτή η υπόθεση.
  More emphatic, like It is this case/assumption that is important or This case is important with stress on important.

So the original sentence is the safest basic pattern for learners.

Greek normally writes an accent mark on the stressed syllable of a word.

In this sentence:

• Αυτή
• υπόθεση
• είναι
• σημαντική

all have a stressed syllable, so they show an accent mark.

But η is a very short unstressed article, so it has no accent.

Be careful not to confuse:

• η = the (feminine singular article)
• ή = or

That accent changes the word.

A simple pronunciation guide is:

• Αυτή = af-TEE
• η = ee
• υπόθεση = ee-PO-thay-see
• είναι = EE-ne
• σημαντική = see-man-dee-KEE

So the whole sentence is approximately:

af-TEE ee-PO-thay-see EE-ne see-man-dee-KEE

A few helpful notes:

• υ here sounds like ee
• θ sounds like English th in think
• αι in είναι sounds like e / eh in Modern Greek, though many beginner guides simplify it differently

Yes, that is possible, but it does not mean exactly the same thing in structure.

• Αυτή η υπόθεση είναι σημαντική
  = This case/assumption is important

• Αυτή είναι σημαντική υπόθεση
  = more like This is an important case/assumption

In the second sentence, αυτή stands more independently as this, and σημαντική υπόθεση functions more like an important case/assumption.

So the original sentence is the better match for This case/assumption is important.

Not always. Υπόθεση can have different meanings depending on context, such as:

• case
• assumption
• hypothesis
• matter/affair

So even if you already know the intended meaning in your lesson, it is useful to remember that the noun is somewhat flexible.

That is very common in Greek vocabulary: one word may cover several related meanings, and context tells you which one is meant.