Le passé composé: the explanation of how to choose the auxiliary verb
The passé composé is the most common French past tense, therefore you'll need it every time you want to write in the past.
Luckily, you only have two auxiliary verbs to pick up from (avoir and être) but how to choose?
Firstly, most of French verbs use the auxiliary verb avoir, whilst the auxiliary être is used in some particular cases.
J'ai fait mes devoirs (I did me homework)
Elles ont gagné le match (They won the game)
But when to use the auxiliary être?
In most of the cases, the auxiliary verb être is used with pronominal verbs (se taire, se laver, se rencontrer, s'aimer...etc) and action verbs (sortir, entrer, aller, revenir, tomber...etc).
Hier, Je ne suis pas allé à l'école (Yesterday, I didn't go to school)
Il s'est lavé les mains avant de manger (He washed his hands before eating)
! With the auxiliary être, the conjugated verb accords with the gender and type of the phrase subject.
Elles sont allées à l'école (They went to school)
Vous vous êtes levés . (You woke up)
! All transitive verbs uses avoir as auxiliary verb. Some verbs that generally use être can be transitive in some cases and the most common example for that is the verbe tomber.
transitive: Il a tombé sa veste (He dropped his jacket)
intransitive: La nuit est tombée (The night fell)
I hope this was helpful.
"Il a tombé sa veste" sounds very colloquial (maybe regional?) to me and a better translation of "he dropped his jacket" would be "il a fait tomber sa veste" ("he made his jacket drop" literally).
A better example might be "il est descendu" (he walked down) vs. "il a descendu l'escalier" (he walked down the stairs) or "elle est rentrée" ("she went in") vs. "elle a rentré la voiture" ("she parked the car inside").