Mi bedaŭras, ke mi ne povis veni al via festo hieraŭ.

Yes. Bedaŭri means to regret or to be sorry about something. So Mi bedaŭras... literally means I regret..., but it is a very natural way to express I'm sorry... in Esperanto.

English often uses an adjective, as in I am sorry, but Esperanto commonly uses a verb here. If you wanted a more direct apology, you might also hear Mi pardonpetas (I apologize / I ask forgiveness), but Mi bedaŭras is perfect for expressing regret.

Because ke introduces a subordinate clause, and Esperanto commonly separates that clause with a comma.

So the sentence is divided like this:

• Mi bedaŭras = main clause
• ke mi ne povis veni al via festo hieraŭ = subordinate clause

The comma helps show that structure clearly.

Ke means that and introduces the content of what is being regretted.

So:

• Mi bedaŭras = I’m sorry / I regret
• ke mi ne povis veni... = that I couldn’t come...

In English, that is often omitted, but in Esperanto ke is normally kept.

Because the inability happened in the past.

Esperanto verb endings show tense very regularly:

• -as = present
• -is = past
• -os = future

So:

• povas = can / am able to
• povis = could / was able to

Since the sentence refers to hieraŭ (yesterday), the past tense is needed: mi ne povis veni = I couldn’t come.

Because povi is followed by another verb in the infinitive.

This works much like English can come, could come, want to come, must come. In Esperanto, after verbs like povi, the second verb stays in the infinitive:

• povas veni = can come
• povis veni = could come
• volas veni = wants to come

So mi ne povis veni literally means I was not able to come.

Because those two mean different things.

• Mi ne povis veni = I couldn’t come / I was unable to come
• Mi povis ne veni = I could choose not to come / I was able not to come

In other words, ne changes meaning depending on what it negates.

Here, ne negates povis, so the meaning is inability, not a choice to stay away.

Because al already shows direction: to your party.

In Esperanto, when a noun follows a preposition like al, it normally does not take -n. So:

• al via festo = to your party

The accusative -n is not used here because al already expresses the destination.

A useful rule of thumb is:

• if al is present, you usually do not add -n to the noun
• -n for direction is more relevant in other kinds of expressions, not after al

Because the clause after ke has its own subject.

The sentence has two clauses:

• Mi bedaŭras
• ke mi ne povis veni al via festo hieraŭ

Even though the subject is the same person in both clauses, Esperanto normally states it again inside the subordinate clause.

If you want to avoid repeating mi, Esperanto can sometimes use an infinitive construction instead:

• Mi bedaŭras ne povi veni al via festo hieraŭ.

That also means I’m sorry I can’t/couldn’t come to your party yesterday, depending on context.

Hieraŭ is a time adverb meaning yesterday. In this sentence, it most naturally refers to the event in the subordinate clause: not being able to come.

So the sense is:

• I’m sorry that yesterday I couldn’t come to your party
• or more naturally in English, I’m sorry I couldn’t come to your party yesterday

Yes, it can move. Esperanto word order is fairly flexible. For example:

• Mi bedaŭras, ke mi hieraŭ ne povis veni al via festo.

That means essentially the same thing.

If you put hieraŭ at the very beginning, it may sound more like the regretting happened yesterday, so placement can slightly affect what it seems to modify.

Ŭ is a very short glide sound, similar to the w part in English now or ow. It does not usually form a syllable by itself.

In these words:

• aŭ sounds roughly like the vowel in now
• bedaŭras is roughly be-DAU-ras
• hieraŭ is roughly hi-E-raŭ

Also remember that Esperanto stress is on the second-to-last syllable:

• beDAŬras
• hiERaŭ

That regular stress rule is one of the nice things about Esperanto pronunciation.