Αυτή η μερίδα είναι πολύ μεγάλη για εμένα, αλλά ο φίλος μου θέλει και δεύτερη.

In Greek, this/that normally goes together with the definite article.

So:

• αυτή = this
• η = the
• μερίδα = portion / serving

Together, Αυτή η μερίδα literally looks like this the portion, but that is the normal Greek way to say this portion.

All three words match in:

• gender: feminine
• number: singular
• case: nominative

You will see this pattern very often:

• αυτό το παιδί = this child
• αυτή η πόρτα = this door
• αυτοί οι φίλοι = these friends

Greek nouns have grammatical gender: masculine, feminine, or neuter.
Μερίδα is a feminine noun, so the words that go with it must also be feminine.

That is why you get:

• αυτή and not αυτός or αυτό
• η and not ο or το
• μεγάλη and not μεγάλος or μεγάλο
• δεύτερη and not δεύτερος or δεύτερο

So even if the meaning is already clear, gender is important because it controls the forms of articles, adjectives, and pronouns around the noun.

Because adjectives in Greek must agree with the noun they describe.

Here, μερίδα is:

• feminine
• singular
• nominative

So the adjective must also be feminine singular nominative:

• μεγάλη = feminine
• μεγάλος = masculine
• μεγάλο = neuter

So:

• η μερίδα είναι μεγάλη = the portion is big
• ο καφές είναι μεγάλος = the coffee is big/large
• το πιάτο είναι μεγάλο = the plate is big

Πολύ means very when it modifies an adjective or adverb.

So:

• πολύ μεγάλη = very big / very large

That is the straightforward grammar here.

In context, πολύ μεγάλη για εμένα can sound like too big for me, even though word-for-word it is closer to very big for me. Greek often uses this kind of wording naturally in situations like food portions.

Yes, you can very commonly say για μένα instead.

Here is what is happening:

• για = for
• εμένα = me (strong form)

The preposition για takes the accusative case, so εμένα is in the accusative.

Both are correct:

• για εμένα
• για μένα

The shorter form για μένα is very common in everyday speech.
The longer form για εμένα can sound a bit fuller or more emphatic.

So:

• Αυτή η μερίδα είναι πολύ μεγάλη για μένα = perfectly natural everyday Greek

In Greek, the unstressed possessive words like μου usually come after the noun.

So:

• ο φίλος μου = my friend
• η αδερφή μου = my sister
• το σπίτι μου = my house

This is one of the big word-order differences from English.

Also notice that Greek normally keeps the article:

• ο φίλος μου literally = the friend my

That is just the normal Greek structure.

It can definitely feel like would like in context.

Literally:

• θέλει = he/she wants

But in everyday Greek, especially when talking about food, drink, or ordering, θέλω can sound very natural where English might prefer:

• want
• would like
• feel like having

So here:

• ο φίλος μου θέλει και δεύτερη
  can mean
• my friend wants a second one
• my friend would like another one

Both reflect the idea well.

Here και does not simply mean the basic and.

In this sentence, και has the sense of:

• also
• even
• another / one more, depending on context

So θέλει και δεύτερη means something like:

• he wants a second one too
• he even wants another one
• he wants a second helping as well

This is a very common use of και in Greek. It often adds the idea of one more or in addition.

The noun is left out because it is already understood from the context.

Earlier in the sentence we have μερίδα. So δεύτερη really means:

• δεύτερη μερίδα = a second portion / a second serving

Greek often omits a repeated noun when it is obvious. English does this too:

• I’ll take the blue shirt, not the red one
• She ordered a coffee, and I ordered another

So here δεύτερη means second one / second serving.

Because the omitted noun is feminine: μερίδα.

Since δεύτερη is describing an understood μερίδα, it must match that noun in gender, number, and case.

So:

• δεύτερη = feminine singular
• understood noun = μερίδα (feminine singular)

If the hidden noun were masculine or neuter, the form would change:

• δεύτερος = masculine
• δεύτερο = neuter

The main subject nouns are in the nominative:

• Αυτή η μερίδα = nominative, because it is the subject of είναι
• ο φίλος μου = nominative, because it is the subject of θέλει

The pronoun after για is in the accusative:

• εμένα = accusative, because it follows the preposition για

So the basic pattern is:

• subject (nominative)
  • verb
• για
  • accusative pronoun

That is a useful structure to remember.