The most common case of verbs that use either être or avoir is verbs that can be either intransitive or transitive. This is the case for monter, passer, retourner, sortir. The auxiliary is avoir when the verb is transitive (the auxiliary is always avoir for transitive verbs, because être + past participle is a passive, not a compound past). The auxiliary is être with an intransitive construction.

Il est monté au troisième étage.   (He went up to the third floor.)
Il a monté l’escalier.   (He climbed the stair.)

There are other cases, generally more subtle. In the case of apparaître, to a first approximation, you can consider that the auxiliary is always être. There can be a distinction between the action of appearing (something became visible in the past → avoir) and the state of appearing (something looked a certain way in the past → être) but the use of avoir is disappearing in late 20th-early 21st century French.

See also this much more detailed answer in French.