Ar trebui să termin raportul până mâine.

Ar trebui să means something like should or ought to.

So:

• Ar trebui să termin raportul până mâine.
• = I should finish the report by tomorrow.

It expresses a recommendation, expectation, or mild obligation, not a very strong command.

The difference is mainly strength and nuance:

• Trebuie să termin raportul până mâine = I must / have to finish the report by tomorrow
• Ar trebui să termin raportul până mâine = I should finish the report by tomorrow

So trebuie is stronger and more direct, while ar trebui is softer.

Grammatically, ar trebui is a conditional form that often gives the meaning of should.

In Romanian, many verbs and verbal expressions are followed by să + verb instead of an infinitive.

So Romanian says:

• ar trebui să termin

not

• ar trebui termina

This is very common in Romanian. Where English often uses to + verb, Romanian frequently uses să + a finite verb.

Because after să, Romanian usually uses a conjugated verb form, not the dictionary infinitive.

Here:

• a termina = to finish (infinitive)
• să termin = that I finish / for me to finish in structure, but in natural English here simply to finish

Since the speaker is I, the verb is in the 1st person singular:

• eu termin = I finish
• să termin = the same person/number after să

So ar trebui să termin literally follows Romanian grammar, even though English would say I should finish.

Romanian often leaves out subject pronouns when they are clear from the verb ending.

Here, termin shows 1st person singular, so the subject is understood as eu = I.

You could say:

• Eu ar trebui să termin raportul până mâine

but normally eu is omitted unless you want emphasis or contrast.

Raportul means the report.

The ending -ul is the Romanian definite article attached to the noun:

• raport = report
• raportul = the report

Unlike English, Romanian usually puts the at the end of the noun rather than as a separate word.

Până mâine means by tomorrow or until tomorrow, depending on context.

In this sentence, the natural English meaning is:

• by tomorrow

because the idea is that the report should be finished no later than tomorrow.

So:

• până = until / by
• mâine = tomorrow

In this sentence, it is best understood as by tomorrow.

That is because the action is something that should be completed before a deadline:

• Ar trebui să termin raportul până mâine.
• = I should finish the report by tomorrow.

If you translate it as until tomorrow, it can sound less natural in English because finish usually goes with a deadline: by tomorrow.

The given word order is natural and neutral:

• Ar trebui să termin raportul până mâine.

Romanian word order is fairly flexible, but changing it can change emphasis.

For example:

• Până mâine ar trebui să termin raportul.
  This emphasizes the deadline by tomorrow.

• Raportul ar trebui să-l termin până mâine.
  This puts more focus on the report, though this structure is a bit more marked.

For a learner, the original sentence is the safest standard version.

It is typically followed by să + a conjugated verb.

Examples:

• Ar trebui să plec. = I should leave.
• Ar trebui să înveți. = You should study.
• Ar trebui să ajungem devreme. = We should arrive early.

So in your sentence:

• Ar trebui să termin...
  follows a very common Romanian pattern.

Yes. Ar trebui să can express:

• advice
• expectation
• mild obligation
• what would be a good idea

So this sentence might mean:

• It would be a good idea for me to finish the report by tomorrow
• I’m expected to finish the report by tomorrow
• I should probably finish the report by tomorrow

The exact nuance depends on context.

Yes. That would also be correct.

The difference is slight:

• a termina = to finish
• a finaliza = to finalize / complete

A finaliza can sound a bit more formal or more professional, so with raportul, it may fit well in work-related contexts.

Only the main verb after să changes to match the subject.

For example:

• Ar trebui să termin raportul până mâine. = I should finish the report by tomorrow.
• Ar trebui să termini raportul până mâine. = You should finish the report by tomorrow.
• Ar trebui să termine raportul până mâine. = He/She should finish the report by tomorrow.
• Ar trebui să terminăm raportul până mâine. = We should finish the report by tomorrow.

So ar trebui să stays the same, and the following verb changes according to the subject.