Το λευκό χαλί ταιριάζει με την πράσινη κουρτίνα.

It is:

subject + verb + prepositional phrase

• Το λευκό χαλί = the subject
• ταιριάζει = the verb
• με την πράσινη κουρτίνα = a phrase meaning with the green curtain

So Greek is using a very natural structure here: The white carpet matches with the green curtain.

Because Greek normally uses the definite article with each definite noun phrase.

So:

• το λευκό χαλί = the white carpet
• την πράσινη κουρτίνα = the green curtain

In English, article use is lighter in some situations, but in Greek it is very normal and expected to include the article here.

Because adjectives in Greek must agree with the noun they describe in gender, number, and case.

The noun χαλί is neuter singular, so the adjective must also be neuter singular:

• masculine: λευκός
• feminine: λευκή
• neuter: λευκό

Since the sentence has το ... χαλί, the correct form is το λευκό χαλί.

For the same reason: adjective agreement.

The noun κουρτίνα is feminine singular, so the adjective must be feminine singular too:

• masculine: πράσινος
• feminine: πράσινη
• neuter: πράσινο

So:

• η πράσινη κουρτίνα = the green curtain
• την πράσινη κουρτίνα = the green curtain, after a preposition here

Because the preposition με normally takes the accusative in Modern Greek.

So the noun phrase changes from nominative to accusative:

• nominative: η πράσινη κουρτίνα
• accusative: την πράσινη κουρτίνα

That is why you see την here.

Because many feminine nouns ending in -α have the same form in the nominative singular and accusative singular.

So:

• nominative: η κουρτίνα
• accusative: την κουρτίνα

The noun ending does not change, but the article does:

• η → την

The adjective πράσινη also happens to look the same in nominative and accusative feminine singular.

Ταιριάζει is the 3rd person singular present form of ταιριάζω.

That matches the singular subject το λευκό χαλί.

A few present-tense forms are:

• ταιριάζω = I match / I fit / I suit
• ταιριάζεις = you match
• ταιριάζει = it matches
• ταιριάζουμε = we match
• ταιριάζουν = they match

Here, the carpet is singular, so ταιριάζει is correct.

Because with the meaning match / go with, Greek normally says ταιριάζω με.

So:

• ταιριάζει με την πράσινη κουρτίνα = it matches the green curtain / it goes with the green curtain

A direct-object structure like ταιριάζει την πράσινη κουρτίνα would not sound right for this meaning.

Yes. Greek word order is more flexible than English because the endings and articles show grammatical relationships.

The sentence as given is a neutral, natural order:

Το λευκό χαλί ταιριάζει με την πράσινη κουρτίνα.

But you could also move things for emphasis, for example:

Με την πράσινη κουρτίνα ταιριάζει το λευκό χαλί.

That puts more focus on with the green curtain.

Yes. In this sentence, το λευκό χαλί is the thing doing the matching, so it is the subject.

A small detail that can confuse learners: with neuter nouns, the nominative and accusative often look the same. So το χαλί could in principle be either form in another sentence. Here, though, the meaning and structure show that it is the subject.

A rough pronunciation guide is:

to lef-KO kha-LEE te-ria-ZI me tin PRA-si-ni koor-TEE-na

A few helpful notes:

• χ in χαλί is not exactly English h. It is a rougher sound, like the ch in German Bach.
• The stress marks show which syllable is stressed:
  • λευκό
  • χαλί
  • ταιριάζει
  • πράσινη
  • κουρτίνα

If you place the stress correctly, your Greek will sound much more natural.

Both the noun phrase and the verb would change.

For example:

• Το λευκό χαλί ταιριάζει... = the white carpet matches...
• Τα λευκά χαλιά ταιριάζουν... = the white carpets match...

So you would see:

• το → τα
• λευκό → λευκά
• χαλί → χαλιά
• ταιριάζει → ταιριάζουν

That is a good example of agreement in Greek.