The traditional definition of when to use essere versus avere as your auxiliary for the present perfect is the following: use essere for intransitive verbs; use avere for transitive verbs.
But as you will see below, it can be very difficult to identify a transitive versus an intransitive verb, so a more useful rule regarding the choice of essere or avere is:
Use essere as your auxiliary verb to form the passato prossimo if:
the verb expresses change in location (andare, partire, arrivare, etc.)
the verb expresses change in state (morire, nascere, diventare, etc.)
the verb expresses staying in location (stare, restare, rimanere, etc.)
If your verb does not express one of these three conditions (and the majority of verbs do not express one of these three conditions) e.g., mangiare (to eat), studiare (to study), comprare (to buy), then it probably takes avere as its auxiliary, and it is probably a transitive verb (meaning it can go with an object noun phrase, e.g., “hanno visto Maria a Fiesole.”)
The following two sections provide tight guidelines regarding the choice of auxiliary verb, but not every case fits neatly and unambiguously into these guidelines. There will be times when you simply must memorize a verb as “a verb that takes essere” or “a verb that takes avere.” Lastly, there are also several verbs that can take either auxiliary, usually with a subtle distinction in meaning.