As Patricia mentions in her recent video, the French conditional mood only comes in two tenses: present and past. While the present conditional expresses something you would do, the past conditional expresses something you would have done. We discussed the present conditional in our previous lesson, so now we'll focus on the past.
The past conditional is a compound tense, which means it's made up of multiple parts. Two parts, to be exact: an auxiliary verb (avoir or être) in the conditional, plus the past participle of the main verb. Here's an example of the verb pouvoir (to be able to) in the past conditional:
On aurait pu les cuire individuellement, mais là ça va le faire.
We could have cooked them individually, but here, this'll do it.
Cap. 49, 4 Mains pour 1 Piano: Médaillon de Homard - Part 2
Like most verbs, pouvoir combines with the auxiliary verb avoir (to have) in compound past tenses. But as Patricia explains in another video, some verbs combine with être (to be) in those instances, such as the verbs aller (to go) and naître (to be born):
Je serais allé à la plage mais il faisait trop froid.
I would have gone to the beach, but it was too cold.
L'histoire officielle dit que ce drapeau serait né sous la Révolution française de dix-sept cent quatre-vingt-neuf.
Official history says that this flag was supposedly born under the French Revolution of seventeen eighty-nine.
Cap. 6-7, Le saviez-vous? - Histoire du drapeau français
The important thing to remember is that in the past tense, you only need to conjugate the auxiliary verb in the conditional, not the main verb (so you wouldn't say on aurait pourrait or je serais irais, for instance).
It's easy to confuse the past conditional with the pluperfect (or plus-que-parfait) tense, which is used to describe things that happened in the remote past. Both constructions contain an auxiliary verb followed by a past participle (in the pluperfect, the auxiliary verb is in the imperfect tense, not the conditional), and you'll often find both of them in sentences containing si (if) clauses:
Hier, j'aurais levé le bras pour appeler le taxi si j'avais d'abord soigné mon épaule.
Yesterday, I would have raised my arm to hail the taxi if I had treated my shoulder first.
Cap. 39-41, Le saviez-vous? - Le mode du conditionnel
We'll talk about si clauses in further detail in a future lesson. In the meantime, you might want to check out the song Si by Zaz, which contains a good number of si clauses and verbs in the conditional.