Auxiliary verbs are also known as helping verbs, because they help form compound conjugations. The key thing to remember about compound conjugations is that it's the auxiliary verb which conjugates for the required tense or mood; the main verb is always a past participle.
When the subject of a French verb is nous, vous, ils, or elles, it's obvious which verb conjugation you need, because those plural pronouns are included in verb tables. But it's a bit trickier with compound subjects made up of multiple names, nouns, and/or pronouns. In these cases, you need to take a moment to figure out which plural subject pronoun those items add up to, so that you know what to conjugate for.
French conditional conjugations are, along with the future, the easiest in the entire language.
The conditional perfect is a compound verb form, which means its conjugation has two components: the auxiliary verb in the conditional plus the past participle of the main verb.
The second form of the conditional perfect suffers from something of an identity crisis: it looks like the pluperfect subjunctive but has the value of the conditional perfect. The second form of the past conditional is used mostly in si clauses and is a literary tense, so is found only in very formal written French.
In the present tense, the irregular -ir verbs devoir (must), pouvoir (can), and vouloir (want) are conjugated according to a similar pattern.
In the present tense, the irregular -re verbs dire (to say, tell), écrire (to write), and lire (to read) are conjugated mostly according to the same pattern.
In the present tense, the irregular -ir verbs dormir (to sleep), partir (to leave), and sortir (to go out) are conjugated according to the same pattern.
Like all compound verb forms, future perfect conjugations have two components.
French future conjugations are, along with the conditional, the easiest in the entire language.