Το αγόρι θέλει παγωτό βανίλια μετά το φαγητό.

Because Greek grammatical gender does not always match natural gender.

αγόρι is a neuter noun, so it takes neuter words with it: το αγόρι, το μικρό αγόρι, το καλό αγόρι.
Even though the person is male, the grammar of the noun is still neuter.

This is something English speakers often need time to get used to: in Greek, noun gender is a grammar category, not just a biological one.

θέλει is the 3rd person singular present form of θέλω, meaning to want.

So:

• θέλω = I want
• θέλεις = you want
• θέλει = he/she/it wants

Here it matches το αγόρι, so it means the boy wants.

Greek often leaves out subject pronouns when they are not needed.

The verb ending already tells you the person and number, and in this sentence the subject is also explicitly stated as το αγόρι. So adding αυτός would usually be unnecessary unless you wanted extra emphasis.

That is why Greek often sounds more compact than English in sentences like this.

Because Greek often omits the article with an indefinite or general direct object, especially with food, drink, and things someone wants, eats, buys, etc.

So θέλει παγωτό βανίλια is natural for he wants vanilla ice cream.

Compare:

• θέλει παγωτό βανίλια = he wants vanilla ice cream
• θέλει ένα παγωτό βανίλια = he wants a vanilla ice cream / one vanilla ice cream
• θέλει το παγωτό βανίλια = he wants the vanilla ice cream, a specific one

In Greek, words for flavor or type often come after the main noun.

So παγωτό βανίλια literally looks like ice cream vanilla, but it means vanilla ice cream.

This is a common Greek pattern. English usually puts this kind of word before the noun, but Greek often puts it after.

In this sentence:

• το αγόρι is the subject, so it is in the nominative
• παγωτό βανίλια is the direct object, so παγωτό is in the accusative
• το φαγητό is also in the accusative because it comes after μετά

A useful thing to know is that many neuter singular nouns have the same form in the nominative and accusative. That is why forms like το αγόρι and το φαγητό do not visibly change.

μετά means after, and in Modern Greek it is commonly followed by the accusative.

So μετά το φαγητό is the normal way to say after the meal / after eating.

English speakers sometimes expect an extra preposition because English says after the meal, but in Greek μετά already does that job by itself here.

It can mean either, depending on context.

By itself, φαγητό often means food.
But in the expression μετά το φαγητό, it usually means after the meal or after eating.

So the exact nuance comes from the whole phrase, not just the single word.

No, Greek word order is more flexible than English word order.

The sentence as written is a very natural, neutral order:

Το αγόρι + θέλει + παγωτό βανίλια + μετά το φαγητό

But Greek can move parts around for emphasis, for example:

Μετά το φαγητό, το αγόρι θέλει παγωτό βανίλια.

That still makes sense, but it puts more focus on after the meal.

A rough pronunciation guide is:

to a-GHO-ri THE-li pa-gho-TO va-NI-lia me-TA to fa-yi-TO

A few helpful notes:

• θ sounds like th in think
• γ before ο is a soft throaty sound, not a hard English g
• αι sounds like e
• η, ι, υ all sound like ee
• the written accent mark shows the stressed syllable

So the stressed parts are:

• αγόρι
• θέλει
• παγωτό
• βανίλια
• μετά
• φαγητό