Το μπουφάν μου είναι βρόμικο, πρέπει να το πλύνω.

The first το is the definite article the.

So:

• το μπουφάν = the jacket
• το μπουφάν μου = my jacket

In Greek, possessive words like μου often appear together with the article, so Greek says something closer to the jacket of mine rather than just my jacket.

Because the short possessive form μου normally comes after the noun in Greek.

So:

• το μπουφάν μου = my jacket
• η τσάντα μου = my bag
• το σπίτι μου = my house

This is very normal Greek word order. If you want extra emphasis, Greek can also use δικός μου / δική μου / δικό μου, but that is not needed here.

Μπουφάν is neuter.

You can tell from the article and adjective:

• το μπουφάν
• βρόμικο

Both show that the noun is neuter.

Also, μπουφάν is a borrowed word and does not change much in form, so the article and adjective are especially helpful for identifying its gender.

Because adjectives in Greek must agree with the noun in gender, number, and case.

Since μπουφάν is neuter singular, the adjective must also be neuter singular:

• masculine: βρόμικος
• feminine: βρόμικη
• neuter: βρόμικο

So:

• το μπουφάν είναι βρόμικο = correct

Πρέπει means something like:

• must
• have to
• need to
• literally, it is necessary

It is an impersonal verb, which means it does not change to match I, you, he, and so on in the usual way here.

So:

• πρέπει να το πλύνω = I have to wash it
• literally: it is necessary that I wash it

Because Modern Greek usually does not use an infinitive the way English does.

English says:

• I have to wash it

Greek uses:

• πρέπει να το πλύνω

So να introduces the verb that follows expressions like πρέπει, θέλω, μπορώ, and many others.

A useful way to think about it is:

• να + verb often does the job that to + verb or a finite clause does in English

This second το is not the article the. It is the object pronoun it.

So the sentence has two different το forms:

• το μπουφάν = the jacket → article
• να το πλύνω = to wash it → pronoun

Same spelling, different function.

Because weak object pronouns in Greek usually come before the verb.

So Greek says:

• να το πλύνω = to wash it

not:

• να πλύνω το

This is very common in Greek:

• θέλω να το δω = I want to see it
• μπορώ να το κάνω = I can do it
• θα το αγοράσω = I will buy it

This is a very important point. The basic verb is πλένω = I wash, but after να, Greek often chooses between two aspects:

• να πλένω = imperfective, ongoing/repeated/habitual
• να πλύνω = perfective, one complete action

Here, the speaker means a single complete action:

• The jacket is dirty, I need to wash it

So να το πλύνω is the natural choice.

In other words:

• πλένω = wash, be washing, wash regularly
• πλύνω = wash it off / wash it once / get it washed as a complete action

Greek usually leaves out subject pronouns when they are clear from the verb ending.

The ending -ω in πλύνω tells you the subject is I.

So:

• πλύνω = I wash
• πρέπει να το πλύνω = I have to wash it

You could add εγώ for emphasis, but it is not necessary.

Yes, absolutely.

Both are natural:

• Το μπουφάν μου είναι βρόμικο, πρέπει να το πλύνω.
• Το μπουφάν μου είναι βρόμικο, πρέπει να πλύνω το μπουφάν μου.

But the version with το is more natural because it avoids repeating το μπουφάν μου. English does the same thing:

• My jacket is dirty. I need to wash it.

No. You could also write it as two sentences:

• Το μπουφάν μου είναι βρόμικο. Πρέπει να το πλύνω.

That is often the cleaner written option.

With the comma, the two parts feel closely connected in one flow of speech. With a full stop, they feel a bit more separate. Both are understandable.