Passé Composé - 'Etre' or 'Avoir'?

In English, in the past tense, we use a combination of an auxillary verb and a main verb in the past participle form. In the example 'I have eaten', 'have' acts as the auxillary verb and 'eaten' is the past participle of the verb 'to eat'. The same construction applies to the French passé composé.
In this case, the auxillary verb is the present, conjugated form of the verb 'être' or 'avoir'. The choice of 'être' or 'avoir' is determined by which verb we are trying to put into the past tense (the past participle).

e.g. Il a mangé un pizza - He has eaten a pizza
('a' is the auxillary of 'avoir' and 'mangé' is the past participle of the verb 'manger')
e.g. Nous sommes arrivés - We have arrived
('avons' is the auxillary of 'être' and 'arrivés' is the past participle of the verb 'arriver')

The majority of verbs take 'avoir' as their auxillary verb in the passé composé, but there are many which take 'être'. There are a couple of tips for remembering the most common times that 'être' is used.

All reflexive verbs take 'être' in the passé composé: e.g. Je me suis lévé(e) - I got up
e.g. Il s'est habillé - He got dressed
Some verbs relating to non-specific movements, such as 'arriver' or 'partir' take 'être'
Some other common verbs take 'être'

A usual mnemonic for remembering some of the common 'être' verbs is: DR & MRS VANDERTRAMP

Devenir
Revenir

(&)

Montrer
Rester
Sortir

Venir
Aller
Naître
Descendre
Entrer
Retourner
Tomber
Retourner
Arriver
Mourir
Partir

Hope this helps!