Ο αδερφός μου θέλει να αφήσει μουστάκι.

Ο is the masculine singular definite article, so yes, it literally means the.

In Greek, nouns often still take the definite article even when they are made definite by a possessive word. So:

• ο αδερφός μου = my brother
• literally: the brother of me

That is completely normal Greek. English usually does not say the my brother, but Greek does use the article here.

In Greek, the weak possessive forms such as μου, σου, του, της, μας, σας, τους usually come after the noun:

• ο αδερφός μου = my brother
• η αδερφή μου = my sister
• το σπίτι μου = my house

So μου here means my, but grammatically it is the same form as the genitive/object clitic μου.

This noun + clitic order is the normal pattern.

Αδερφός is in the nominative singular masculine.

That is because it is the subject of the sentence: the brother is the one who wants something.

You can see that in the article too:

• ο αδερφός = nominative
• if it were object form, you would expect τον αδερφό

Yes, both mean brother.

A learner may see both forms:

• αδερφός — very common in everyday Modern Greek
• αδελφός — more formal, more conservative, sometimes more bookish

In ordinary conversation, αδερφός is extremely common and natural.

Modern Greek normally does not use an infinitive the way English does.

So where English says:

• He wants to leave
• He wants to grow a moustache

Greek usually says:

• θέλει να...

So θέλει να αφήσει is the normal way to say wants to leave / let / grow depending on context.

You can think of να as the particle that introduces the dependent verb after words like:

• θέλω = I want
• μπορώ = I can
• πρέπει = I must

It looks like the form used in the future, but here it is used after να.

In many traditional explanations, this is called the aorist subjunctive. In more modern descriptions, it is the perfective non-past form.

The important thing for a learner is this:

• να αφήσει presents the action as a single whole result: to let/grow a moustache
• after θέλει, this is the normal choice here

If you used να αφήνει, that would be the imperfective form and would suggest something more ongoing, repeated, or habitual, which does not fit as naturally in this sentence.

So:

• θέλει να αφήσει μουστάκι = he wants to grow a moustache

Yes, αφήνω often means leave, let, or allow, but with hair and facial hair it has a special very common use:

• αφήνω γένια = I grow / let my beard grow
• αφήνω μουστάκι = I grow / let a moustache grow
• αφήνω μαλλιά = I grow my hair

So the idea is literally something like let it grow, and in natural English we translate that as grow.

Greek does have words that can correspond to a/an:

• ένας for masculine
• μία / μια for feminine
• ένα for neuter

But Greek does not use them in all the same places where English uses a/an.

In this expression, the bare noun is the natural choice:

• να αφήσει μουστάκι

That is idiomatic Greek for to grow a moustache.

If you said ένα μουστάκι, it would sound less natural here, unless you were emphasizing a particular kind, style, or some special moustache.

Μουστάκι is a neuter singular noun.

Its basic form is:

• το μουστάκι = the moustache

Here it stays as μουστάκι because it is the direct object in a context where Greek uses the noun without the article.

So even though it is the object of the verb, the form you see is still the standard singular form for this neuter noun.

Yes, Greek word order is more flexible than English word order, but the sentence as written is the most neutral and natural:

• Ο αδερφός μου θέλει να αφήσει μουστάκι.

You can move things around for emphasis, but not every rearrangement sounds equally natural. For a learner, this version is the best one to remember.

A useful general point is:

• Greek word endings and articles often show grammatical function
• English relies more heavily on word order

So Greek can move words more freely, but the neutral order is still the safest choice.

A rough pronunciation guide is:

o a-ther-FOS moo THE-li na a-FI-si moo-STA-ki

A few notes:

• δ sounds like the th in this
• θ sounds like the th in think
• stress falls on:
  • αδερφός
  • θέλει
  • αφήσει
  • μουστάκι

So the rhythm is:

Ο αδερφός μου θέλει να αφήσει μουστάκι.