Το ένα κορδόνι από το παπούτσι μου λύθηκε στον δρόμο, γι’ αυτό σταμάτησα να το δέσω.

Το ένα κορδόνι is more specific than ένα κορδόνι.

• ένα κορδόνι = a shoelace / one shoelace
• το ένα κορδόνι = the one shoelace / one of the shoelaces

In Greek, the article + ένας/μία/ένα is very common when you mean a particular one. Here it suggests one specific lace, probably in contrast with the other one.

By itself, κορδόνι can mean lace, cord, string, shoelace, depending on context.

Because the sentence says από το παπούτσι μου (from my shoe), the meaning is clearly shoelace here.

Από το παπούτσι μου literally means from my shoe, and in context it identifies which lace we are talking about: the lace attached to / belonging to the shoe.

Greek can also say:

• το κορδόνι του παπουτσιού μου = the lace of my shoe
• το κορδόνι από το παπούτσι μου = the lace from my shoe

The version with από feels a bit more conversational and concrete, almost as if you are pointing to the lace coming from the shoe.

Λύθηκε is the aorist of λύνομαι.

Here it means:

• it came undone
• it got untied
• it came loose

Greek often uses a middle/passive-looking form for things that happen to themselves or without focusing on an agent. So το κορδόνι λύθηκε is very natural for the lace came untied.

Yes.

• έλυσε = he/she/it untied something
• λύθηκε = it became untied / it came undone

So:

• Έλυσα το κορδόνι = I untied the lace
• Το κορδόνι λύθηκε = The lace came untied

That active/passive contrast is important.

Yes. Στον is the contraction of σε + τον.

So:

• σε τον δρόμο → στον δρόμο

It means on the road / in the street / while out on the road, depending on context.

You may also see στο δρόμο in modern Greek, because the final -ν is sometimes dropped in writing and speech. Both are common.

Γι’ αυτό means for that reason, because of that, or simply so / that’s why.

It comes from για αυτό, with the final -α of για dropped before the vowel in αυτό. The apostrophe shows that omission:

• για αυτό → γι’ αυτό

This shortened form is very common.

Here it means I stopped in order to tie it.

The idea is:

• the person was walking,
• the lace came untied,
• so they stopped and tied it.

In this kind of sentence, Greek can leave the interrupted action unstated because the context makes it obvious.

If you wanted to make the purpose even more explicit, you could say:

• σταμάτησα για να το δέσω = I stopped in order to tie it

After να, Greek uses a subjunctive form, not an infinitive.

So:

• να δέσω = to tie / that I tie as one complete action
• δένω = present form, I tie / I am tying
• έδεσα = past indicative, I tied

Greek does not use an English-style infinitive here. So να δέσω is the normal structure.

Also, the aorist form δέσω fits because tying the lace is seen as one completed action.

Το refers back to το ένα κορδόνι.

Because κορδόνι is neuter singular, the object pronoun is also neuter singular:

• το κορδόνι → το

So να το δέσω means to tie it.

In Modern Greek, weak object pronouns usually come before the verb:

• το δένω = I tie it
• να το δέσω = to tie it

That is the normal placement for short object pronouns like το, τον, την, μου, σου.

That is the normal Greek way to express possession with these short pronouns.

• το παπούτσι μου = my shoe
• literally, something like the shoe my

Greek usually keeps the article too, so το παπούτσι μου is more natural than just παπούτσι μου in a neutral sentence.

Yes. A few very natural alternatives would be:

• Το ένα κορδόνι του παπουτσιού μου λύθηκε στον δρόμο, γι’ αυτό σταμάτησα να το δέσω.
• Μου λύθηκε το ένα κορδόνι στον δρόμο, γι’ αυτό σταμάτησα να το δέσω.

The basic grammar and meaning stay the same; the differences are mainly about style and emphasis.