Learn Hardcore Greek: Εκείνος μιλάει ελληνικά πολύ καλά.

In modern Greek, αυτός is the normal everyday pronoun for he.
εκείνος is more distant (physically or mentally) and often a bit more formal or emphatic, like that one (as opposed to another one).

So:

Αυτός μιλάει ελληνικά πολύ καλά. = He speaks Greek very well. (neutral)
Εκείνος μιλάει ελληνικά πολύ καλά. = That man / that one speaks Greek very well. (more contrast or distance)

Grammatically, εκείνος here is masculine nominative singular, functioning as the subject of the verb.

Can I leave out εκείνος and just say Μιλάει ελληνικά πολύ καλά?

Yes. Greek is a “pro‑drop” language: subject pronouns are often omitted because the verb ending already shows who the subject is (at least in context).

Μιλάει ελληνικά πολύ καλά. = He/She/It speaks Greek very well.
Context would tell you whether it’s he, she, or it.

Adding εκείνος (or αυτός) usually:

adds emphasis to the subject, or
helps clarify who you mean if it isn’t obvious from context.

So all of these are possible, with slight differences in focus:

Μιλάει ελληνικά πολύ καλά. – Speaks Greek very well. (neutral, context decides who)
Αυτός μιλάει ελληνικά πολύ καλά. – He speaks Greek very well.
Εκείνος μιλάει ελληνικά πολύ καλά. – That one / that man speaks Greek very well, often in contrast to others.

What form of the verb is μιλάει, and what does it correspond to in English?

Μιλάει is:

person: 3rd person singular
tense: present
mood: indicative
verb: μιλάω / μιλώ (to speak)

So μιλάει means he/she/it speaks or he/she/it is speaking.
Modern Greek present tense normally covers both English:

simple present (He speaks Greek) and
present continuous (He is speaking Greek).

So the same Greek sentence Εκείνος μιλάει ελληνικά πολύ καλά can mean either:

He speaks Greek very well (in general), or
He is speaking Greek very well (right now), depending on context.

I sometimes see μιλά instead of μιλάει. Are they the same, and which should I use?

Yes, μιλά and μιλάει are two acceptable spellings of the same form (3rd person singular present).

The verb μιλάω / μιλώ (to speak) has two stylistic patterns:

Everyday/demotic style (very common in speech and modern writing):

εγώ μιλάω – I speak
εσύ μιλάς – you speak
αυτός/αυτή/αυτό μιλάει or μιλά – he/she/it speaks
εμείς μιλάμε – we speak
εσείς μιλάτε – you (pl/polite) speak
αυτοί/αυτές/αυτά μιλάνε – they speak

More formal or older style (you will see it in books, news, more formal writing):

εγώ μιλώ
εσύ μιλάς
αυτός/αυτή/αυτό μιλά
εμείς μιλούμε
εσείς μιλάτε
αυτοί/αυτές/αυτά μιλούν(ε)

Notes:

In speech, μιλά and μιλάει are usually pronounced the same (essentially mi‑LÁ).
As a learner, it’s safe and natural to use the everyday pattern: μιλάω, μιλάς, μιλάει, μιλάμε, μιλάτε, μιλάνε.
You will often see both μιλά and μιλάει in real texts; they mean the same thing in this context.

Why is ελληνικά in the neuter plural? Why not ελληνική γλώσσα?

In Greek, names of languages are usually the neuter plural of the corresponding adjective of nationality:

ελληνικός, ελληνική, ελληνικό → ελληνικά = Greek (language)
αγγλικός, αγγλική, αγγλικό → αγγλικά = English
γαλλικός, γαλλική, γαλλικό → γαλλικά = French
γερμανικός → γερμανικά, ιταλικός → ιταλικά, etc.

Historically, ελληνικά meant Greek things / Greek words, and it has come to mean the Greek language.

So:

Μιλάει ελληνικά. = He/She speaks Greek.
If you really want to say Greek language, you can say η ελληνική γλώσσα, but after verbs like μιλάω (speak), ξέρω (know), καταλαβαίνω (understand), Greeks almost always just use the neuter plural form:
- Ξέρω πολύ καλά ελληνικά. – I know Greek very well.

Should ελληνικά be capitalized, like Greek in English?

No. In Greek, adjectives are not capitalized, and language names are adjectives used as nouns. So:

ελληνικά (Greek, as a language): not capitalized
αγγλικά, γαλλικά, γερμανικά: not capitalized either

What is capitalized:

Names of people and peoples, e.g.
- Έλληνας (a Greek man)
- Ελληνίδα (a Greek woman)
- Έλληνες (Greeks, as a people)

So:

Μιλάει ελληνικά πολύ καλά. – correct
Μιλάει Ελληνικά πολύ καλά. – normally considered wrong in standard spelling (unless at the start of a sentence).

Why do we say πολύ καλά and not πολύ καλό here?

Because we need an adverb, not an adjective.

καλά is an adverb, meaning well.
- Μιλάει καλά. – He/She speaks well.
καλός / καλή / καλό are adjectives, meaning good (masculine / feminine / neuter).
- καλός άνθρωπος – a good man/person
- καλό παιδί – a good child

After a verb of action like μιλάει (speaks), Greek uses an adverb to say how (in what manner) something is done:

Μιλάει καλά. – He speaks well.
Μιλάει πολύ καλά. – He speaks very well.

πολύ here is an adverb meaning very, modifying καλά.
πολύ καλό would be very good (neuter adjective) and would normally modify a neuter noun, e.g. πολύ καλό βιβλίο (a very good book), not a verb.

What is the difference between πολύ, πολλή, πολλοί, πολλά, etc., and why is it πολύ here?

There are two uses:

Adverb πολύ = very / a lot
- Used before adjectives and adverbs.
- Does not change form (no gender/number).
- Examples:
  - πολύ καλά – very well
  - πολύ καλός – very good (masc.)
  - πολύ ωραίο – very nice (neuter)
- In our sentence, πολύ modifies the adverb καλά, so it is the invariable adverb.
Adjective πολύς, πολλή, πολύ = much / many / a lot of
- Used before nouns, and it agrees with the noun in gender, number, and case.
- Singular:
  - πολύς χρόνος – much time (masc.)
  - πολλή δουλειά – a lot of work (fem.)
  - πολύ νερό – a lot of water (neuter)
- Plural:
  - πολλοί άνθρωποι – many people (masc.)
  - πολλές γυναίκες – many women (fem.)
  - πολλά βιβλία – many books (neuter)

In Εκείνος μιλάει ελληνικά πολύ καλά, πολύ is the adverb (very), not the adjective (many/much), so it stays as πολύ.

Could I change the word order, for example Εκείνος μιλάει πολύ καλά ελληνικά or Πολύ καλά μιλάει ελληνικά?

Yes. Greek word order is fairly flexible, especially with adverbs. All of these are grammatical:

Εκείνος μιλάει ελληνικά πολύ καλά.
Εκείνος μιλάει πολύ καλά ελληνικά.
Μιλάει ελληνικά πολύ καλά.
Μιλάει πολύ καλά ελληνικά.
Πολύ καλά μιλάει ελληνικά (εκείνος).

All mean roughly He speaks Greek very well, but the focus can shift slightly:

Placing πολύ καλά earlier (e.g. Πολύ καλά μιλάει ελληνικά) tends to emphasize how well he speaks, sometimes with a contrast:
- Πολύ καλά μιλάει ελληνικά, αλλά δεν γράφει τόσο καλά.
  – He speaks Greek very well, but he doesn’t write it so well.
Moving ελληνικά closer to μιλάει is also common:
- Μιλάει πολύ καλά ελληνικά. – Very natural.

As a learner, the two most neutral and common variants you can copy are:

Μιλάει πολύ καλά ελληνικά.
Μιλάει ελληνικά πολύ καλά.

How would I say She speaks Greek very well and They speak Greek very well using the same pattern?

Using αυτός / αυτή / αυτοί (more common in speech than εκείνος):

She speaks Greek very well.
- Αυτή μιλάει ελληνικά πολύ καλά.
- Or simply Μιλάει ελληνικά πολύ καλά. – context would show it’s she.
They speak Greek very well. (masculine or mixed group)
- Αυτοί μιλάνε ελληνικά πολύ καλά.
- Or simply Μιλάνε ελληνικά πολύ καλά.

For completeness:

Feminine plural they: Αυτές μιλάνε ελληνικά πολύ καλά.
Neuter plural they (things / children / groups seen as neuter): Αυτά μιλάνε ελληνικά πολύ καλά.

Verb change summary:

μιλάει – he/she/it speaks
μιλάνε (or more formal μιλούν(ε)) – they speak

What do the accent marks in εκείνος, μιλάει, ελληνικά, πολύ, καλά tell me about pronunciation?

Modern Greek uses a single accent mark (΄) to show which syllable is stressed. Every word of two or more syllables normally has one accent.

In our sentence:

εκείνος → ε‑ΚΕΙ‑νος → stress on εί: εκείνος (e‑KÍ‑nos)
μιλάει → μι‑ΛΑ‑ει → stress on λά: μιλάει (mi‑LÁ‑e; often heard as just mi‑LÁ)
ελληνικά → ε‑λλη‑νι‑ΚΑ → stress on κά: ελληνικά (e‑lli‑ni‑KÁ)
πολύ → πο‑ΛΥ → stress on λυ: πολύ (po‑LÍ)
καλά → κα‑ΛΑ → stress on λά: καλά (ka‑LÁ)

A few extra points useful for this sentence:

The letters η, ι, υ, ει, οι, υι are all pronounced like i in machine.
- So μιλάει is pronounced something like mi‑LÁ‑e (and often contracted towards mi‑LÁ in fast speech).
- ελληνικά has two i sounds: e‑llee‑ni‑KÁ.
The accent does not change the sound of the vowel, only which syllable is stressed.

So the accent marks tell you where to put the stress, which is very important for natural Greek pronunciation.