An auxiliary verb is a conjugated verb used in front of another verb in compound tenses in order to indicate the mood and tense of the verb.
In French, the auxiliary verb is either avoir or être.
All French verbs are classified by which auxiliary verb they take, and they use the same auxiliary verb in all compound tenses.
Most French verbs use avoir.
The following is a list of verbs (and their derivatives) that require être (I did not provide meanings for the derivatives):
aller - to go
arriver - to arrivedescendre (redescendre) - to descend / go downstairsentrer (rentrer) - to entermonter (remonter) - to climb
mourir - to die
naître (renaître) - to be born
partir (repartir) - to leave
passer - to pass, spend timerester - to stayretourner - to return
sortir (ressortir) - to go outtomber (retomber) - to fall
venir (devenir, parvenir, revenir) - to come
These are all verbs of a certain kind of movement. You do get used to these verbs over time and one day you'll just know whether to use être or avoir without even having to think about it. In the meantime, though, you might find it useful to use a mnemonic device.
Notes1. All pronominal verbs use être as the auxiliary verb as well:
se baigner - to bathe
s'habiller - to get dressed
s'imaginer - to imaginese
laver - to wash
se lever - to get up
se moquer - to make fun of
se préparer - to prepare oneself
se promener - to go for a walk
se raser - to shave
se reposer - to rest etc.
2. For all verbs conjugated with être in all of the compound tenses, the past participle has to agree with the subject in gender and number (learn more):
Il est allé - Elle est allée - Ils sont allés - Elles sont allées
3. Verbs are conjugated with être only when they are intransitive.
When the above verbs are used transitively, avoir is used as the auxiliary verb.
Je suis sorti - I went out.
J'ai sorti la voiture - I took the car out.
Il est descendu - He went downstairs.
Il a descendu la valise - He took the suitcase down.