Descendre 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 descendre 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 + descendu [de, sur, etc]

= to get off [something] 
= to get out of  [something/somewhere] 
= to go/come down from/to/into [something/somewhere]
 
Je suis descendu du train avant elle.I got off the train before her.

Tu es descendu de la voiture.You got out of the car.

Ils sont descendus de Paris pour le mariage.They came down from Paris for the wedding.

Elle est descendue à la cave chercher une bouteille de vin.She went down to the basement to get a bottle of wine.

Note that in each case where être is the auxilliary, the verb descendre is followed by a preposition (en, sur, de, dans, à etc.). 
In these cases descendre is usually about getting off [something], getting out of [something] or coming down from [somewhere].

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

avoir + descendu [quelque chose] ou [quelqu'un]

= to go/come/climb down [something] 
= to take [something] down -> physically move [something] to a lower position 
= to take [someone / animated being] down -> to shoot down, kill
 
J'ai descendu les escaliers aussi vite que j'ai pu.I went down the stairs as fast as I could.

Il a descendu le Parrain.He took down the Godfather (!!)

J'ai descendu les boîtes au sous-sol.I took the boxes down to the basement.

Tu as descendu le cadeau de Pierre?You took Pierre's present downstairs?
-> Here it literally means You took Pierre's present down (somewhere), which in a geographical context implies down a physical level, such as down some stairs or in the elevator. Therefore in English, you would translate it as downstairs when no clear destination (e.g. to the basement) is expressed. 

When descendre 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 descendre means in English. English verbs are very often 'prepositional', meaning we say things like to get off a plane as well as disembark a boat which are equivalent in meaning but grammatically very different - our verbs very often have prepositions where they don't in French! So, ask yourself if there's a preposition in the French, not the English! 
 
Here is the list of all "two-auxiliaryverbs in compound tenses:
 

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Examples and resources

J'ai descendu les boîtes au sous-sol.I took the boxes down to the basement.
Ils sont descendus de Paris pour le mariage.They came down from Paris for the wedding.
Elle est descendue à la cave chercher une bouteille de vin.She went down to the basement to get a bottle of wine.

avoir + descendu


J'ai descendu les escaliers aussi vite que j'ai pu.I went down the stairs as fast as I could.
Tu as descendu le cadeau de Pierre?You took Pierre's present downstairs?
Il a descendu le Parrain.He took down the Godfather (!!)

être + descendu


Je suis descendu du train avant elle.I got off the train before her.
Tu es descendu de la voiture.You got out of the car.
Jacques est descendu du haricot magique.Jack got off the beanstalk.
Clever stuff underway!