Learn French: Ils nous ont assuré que la révision aurait montré l'avancement. - They assured us that the revision would have shown the progress.

This sentence uses reported speech. If they were speaking directly, they would have used the future perfect: "La révision aura montré l'avancement" (The revision will have shown the progress). But because the main reporting verb (ont assuré) is in the past, the future perfect must "backshift" into the past conditional (aurait montré).

How exactly is the past conditional aurait montré formed in this sentence?

It is made of two parts, similar to English. You take the auxiliary verb (here, avoir) and conjugate it in the present conditional to get aurait (would have). Then, you add the past participle of your main verb, which is montré (shown).

Could I use le progrès instead of l'avancement to mean "the progress"?

Yes, but they carry slightly different nuances. L'avancement refers specifically to the physical progression, status, or headway of a project or task. Le progrès (often used in the plural as les progrès) implies a more general improvement, like making progress in learning a language.

Does the past conditional aurait montré mean the revision definitely showed the progress?

Not necessarily. It only expresses what they claimed would have happened by a certain point in time, from their past perspective. It describes a theoretical completed action, but the sentence alone doesn't confirm if the revision actually took place or succeeded.