How do I know what auxiliary verb to use in the passé composé?

The passé composé is used to describe a single, completed action in the past. The passé composé is a compound tense, meaning you need two components to conjugate the verb: 1) the auxiliary verb (avoir or être) conjugated in the present tense and 2) the past participle of the verb you want to conjugate.
Most verbs construct the passé composé with avoir. For example: il a rangé sa chambre (he tidied his bedroom)
However, there are some few common exceptions that take être instead. An easy way to remember some of these verbs is to use the helpful mnemonic : MRS VANDERTRAMP
MonterRetournerSortir
VenirAllerNaîtreDescendreEntrer ResterTomberRentrerArriverMourirPartir
All of these verbs and their compound forms (e.g. devenir, revenir) take être as their auxiliary verb, e.g. je suis arrivé à la gare (I arrived at the station).
Occasionally, Mrs Vandertramp verbs can take a direct object and so are used transitively. In this case, they will be conjugated with avoir. e.g. J'ai sorti les poubelles
Similarly, all reflexive verbs take être as their auxiliary. Don't forget the reflexive pronoun!e.g. Elle s'est lavée.
Don't forget for all these verbs that take être as their auxiliary verb, the past participle must agree in gender and number with the subject of the sentence. e.g. Ils sont allés dans leur bureaux.