Monter can be used with avoir or être in Le Passé Composé... and changes meaning

Most verbs use either avoir or être as the auxiliary verb in Passé composé (or other compound tense)but monter uses both, depending on its grammatical usage* and what it means in the sentence.
 
*Grammaphile's Corner : the technical grammatical distinction between these cases is actually whether the verb is used in a transitive or intransitive manner. 
- The transitive version (the version with a direct object) uses avoir.
- The intransitive version (lacking a direct object), uses être.
 

être + monté [dans, sur, en, etc.]

= to go inside [something] 
= to get on [something] 
= to embark

On est montés dans la voiture après vous.We went inside the car after you.

Tu es monté sur le toit.You got on the roof.

Nous sommes montés dans l'avion à 6 heures du matin.We got on the plane at 6 am.

Nous sommes tous montés en voiture.We all got in the car.

Note that in each case where être is the auxilliary, the verb monter is followed by a preposition (en, sur, dans, à, ...).   
In these cases, monter is usually about getting inside [something], or going on top of [something].

(See also Agreeing past participle with subject's gender and number with (+ être) verbs in Le Passé Composé)

avoir + monté [quelque chose]

= to go up [something] 
= to take [something] up
= to put [something] together
= to mount [something]

Tu as monté les escaliers en silence.You went up the stairs silently.

Les explorateurs ont monté la colline.The explorers went up the hill.

As-tu monté les valises?Did you take the luggage upstairs?

J'ai monté mon cheval.I mounted my horse.
 

When monter is followed immediately by a noun (as opposed to a preposition), it uses avoir as the auxiliary, like most verbs.
 
It can be very tricky to get the distinction here if you think in terms of what monter means in English. English verbs are very often 'prepositional', meaning we say things like to climb on a horse as well as mount a horse which are equivalent in meaning but grammatically very different - our verbs very often have prepositions where they don't in French!
J'ai monté mon cheval.I mounted my horse.
BUT
Je suis monté à cheval.I rode on horseback.

Here is the list of all "two-auxiliaryverbs in compound tenses:
 

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Examples and resources

Les explorateurs ont monté la colline.The explorers went up the hill.
Je suis monté à cheval.I rode on horseback.
Tu as monté les escaliers en silence.You went up the stairs silently.
Nous sommes montés dans l'avion à 6 heures du matin.We got on the plane at 6 am.
As-tu monté les valises?Did you take the luggage upstairs?
On est montés dans la voiture après vous.We went inside the car after you.
Nous avons monté le lit du bébé nous-mêmes.We put together the baby's bed ourselves.
Nous sommes tous montés en voiture.We all got in the car.
J'ai monté mon cheval.I mounted my horse.
Tu es monté sur le toit.You got on the roof.
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