Το φερμουάρ της τσάντας μου χάλασε και έβαλα τα κέρματα στην τσέπη του παλτού μου.

Greek usually shows of-relationships with the genitive case, not with a separate preposition.

So:

• το φερμουάρ της τσάντας μου = the zipper of my bag
• η τσέπη του παλτού μου = the pocket of my coat

The words της τσάντας and του παλτού are in the genitive, and that is what gives the meaning of the bag and of the coat.

In Greek, the weak possessive forms such as μου, σου, του, της usually come after the noun.

So Greek says:

• η τσάντα μου = my bag
• το παλτό μου = my coat

This is normal Greek word order. English puts my before the noun, but Greek often uses this short genitive pronoun after it.

Greek uses the definite article much more often than English does, including with possessed nouns.

So Greek normally says:

• η τσάντα μου
• το παλτό μου
• η τσέπη του παλτού μου

Even when English would just say my bag or my coat, Greek usually keeps the article.

Because it is in the genitive singular.

The basic form is:

• η τσάντα = nominative, the bag

In the genitive singular it becomes:

• της τσάντας = of the bag

This is a very common pattern for many feminine nouns ending in -α.

Because here too the noun is in the genitive singular.

The basic form is:

• το παλτό = the coat

But after τσέπη we need of the coat, so Greek uses:

• του παλτού

Many neuter nouns have a genitive form in -ού, so this is a useful pattern to remember.

στην is the contracted form of σε + την.

So:

• σε την → στην

Here it means in or into, depending on context:

• στην τσέπη = in the pocket / into the pocket

With a verb like έβαλα (I put), English often prefers into, but Greek still commonly uses σε / στην.

Because after the preposition σε, Greek uses the accusative case.

So the noun phrase is:

• η τσέπη = nominative, basic dictionary form
• την τσέπη = accusative

Since the sentence has στην τσέπη (σε + την τσέπη), the accusative is required.

Both are in the aorist, which is the normal Greek tense for a completed past action.

• χάλασε = it broke / got damaged
• έβαλα = I put

The aorist is often the best match for the English simple past when you are talking about single completed events.

Because βάζω is the present-tense form, while έβαλα is its aorist form.

This verb is not fully regular in the past:

• βάζω = I put / I am putting
• έβαλα = I put in the sense of a completed past action

So this is something you simply have to learn as a verb pair. Greek often has a present stem and a different aorist stem.

In Greek, χαλάω / χαλώ can work in a way that means break, go bad, stop working, or get damaged without needing a passive form.

So:

• Το φερμουάρ χάλασε = The zipper broke / got damaged

This is very natural Greek. It does not mean that someone necessarily broke it on purpose; it just means that the zipper ended up damaged or unusable.

Because it refers separately to the bag and the coat.

• της τσάντας μου = of my bag
• του παλτού μου = of my coat

Greek normally repeats μου when it belongs to two different nouns like this. If you left out the second μου, the sentence would no longer explicitly say that the coat is yours too.

Greek word order is more flexible than English word order, but the sentence as given is very natural and neutral.

This version:

• Το φερμουάρ της τσάντας μου χάλασε και έβαλα τα κέρματα στην τσέπη του παλτού μου.

sounds like a straightforward narrative: first the zipper broke, then I put the coins in my coat pocket.

You could change the order for emphasis, but the original is a very standard way to say it.