Νομίζω ότι το παιδί μοιάζει στη μαμά του όταν χαμογελάει, αλλά στον μπαμπά του όταν θυμώνει.

Ότι introduces the clause that follows Νομίζω. It works like English that in I think that....

So:

• Νομίζω = I think
• Νομίζω ότι... = I think that...

In English, that is often omitted. In Greek, ότι is very common here. You may also see πως in the same role.

Because μοιάζω is normally used with σε + noun/pronoun.

A very useful pattern to learn is:

• μοιάζω σε κάποιον = I look like / resemble someone

So Greek does not structure this like English resemble someone. It uses a prepositional phrase instead.

That is why the sentence has:

• στη μαμά του
• στον μπαμπά του

rather than a direct object.

They are contractions of σε plus the definite article:

• σε + τη(ν) = στη
• σε + τον = στον

So:

• στη μαμά = to/like the mother in form, but here simply like the mother
• στον μπαμπά = to/like the father in form, but again this is just the pattern required by μοιάζω

The difference between στη and στον is due to gender:

• μαμά is feminine
• μπαμπάς is masculine

Του here is the possessive clitic meaning his / her / its, depending on context.

So:

• η μαμά του = his/her mother
• ο μπαμπάς του = his/her father

Greek usually places this possessive word after the noun, not before it as English does.

In this sentence, του refers back to το παιδί.

Not necessarily.

This is a very common point of confusion. The noun το παιδί is grammatically neuter, so Greek naturally uses του to refer back to it, even if the child is biologically male or female.

So:

• το παιδί ... η μαμά του can mean the child ... his mother or the child ... her mother, depending on context.

Greek is following the grammatical gender of παιδί, not necessarily the real-life sex of the child.

Because Greek uses the definite article much more often than English does.

With specific people, family roles, and many ordinary nouns, the article is usually required:

• η μαμά του
• ο μπαμπάς του

After σε, these become:

• στη μαμά του
• στον μπαμπά του

In English, we often say his mother without the. In Greek, the article is still normally there.

Because the noun is not in the nominative form here.

The basic form is:

• ο μπαμπάς = the dad

But after σε (here contracted as στον), Greek uses the accusative:

• τον μπαμπά
• στον μπαμπά

So:

• ο μπαμπάς = nominative
• τον μπαμπά = accusative

By contrast, μαμά does not change form here:

• η μαμά
• τη μαμά

So the feminine noun looks the same, while the masculine one changes.

Normally, it is understood to be the child.

So the sentence means:

• the child looks like the mother when the child smiles
• but like the father when the child gets angry

In Greek, if no new subject is introduced, the subject of the subordinate clause is usually understood to be the same as the main subject. If the mother or father were meant to be the one smiling or getting angry, Greek would normally make that clearer.

Because Greek often omits repeated words when they are easy to understand from context.

The full version could be:

• Νομίζω ότι το παιδί μοιάζει στη μαμά του όταν χαμογελάει, αλλά μοιάζει στον μπαμπά του όταν θυμώνει.

But repeating μοιάζει is not necessary. Greek often leaves out the second occurrence to avoid sounding repetitive.

This kind of omission is very natural.

Both are present tense, third person singular:

• χαμογελάει = he/she smiles
• θυμώνει = he/she gets angry / is getting angry

With όταν, the present tense often describes a general or repeated situation:

• όταν χαμογελάει = when / whenever he smiles
• όταν θυμώνει = when / whenever he gets angry

So the sentence is not necessarily about one single moment happening right now. It is describing a general pattern.

Also, θυμώνει often has the sense of gets angry rather than simply is angry.

Yes. In modern Greek, both χαμογελάει and χαμογελά are common ways to say he/she smiles.

So you may see either:

• όταν χαμογελάει
• όταν χαμογελά

Both are normal.

Just do not confuse χαμογελά with the imperative χαμογέλα (smile!). The accent is different, and so is the meaning.