Perfect tense with 'sein' for staying, being and happening
Most German verbs form the perfect tense with haben, and verbs of movement like gehen or fahren use sein. A small group of verbs does not fit neatly into either box. They describe no movement at all, yet they still take sein: sein itself, bleiben, and verbs of things happening such as passieren and geschehen. So it is ich bin geblieben, not ich habe geblieben.
English speakers trip over this because English uses have for every perfect: I have been, I have stayed, it has happened. Translating word for word gives you haben, which sounds clearly wrong to German ears with these verbs.
The construction is the same as for motion verbs: a form of sein plus the past participle at the end of the clause. Only the choice of helper verb is different from what you might expect.
A useful rule of thumb: if the verb is about remaining in a state (sein, bleiben) or about something occurring or turning out a certain way (passieren, geschehen, erscheinen, scheitern), expect sein. Most of these verbs also have no direct object, so there is nothing that is being done to anything. That is a typical feature of sein verbs.
The most frequent case is sein itself: its perfect is ist gewesen. In everyday speech many people use the simple past war instead, but you will still hear gewesen constantly. Bleiben works the same way.
Passieren and geschehen both mean to happen. Passieren is the everyday word, and geschehen sounds a little more formal or literary. Both take sein, and the person affected goes in the dative.
Erscheinen (to appear, also used for books and films being published or released) and scheitern (to fail) describe how something comes about or turns out. Both take sein. The same goes for misslingen and its opposite gelingen.
Folgen takes the dative and forms its perfect with sein when it means to go after someone.
Not every verb about events takes sein. Stattfinden (to take place) is a common trap: it uses haben.
Folgen in the sense of to obey uses haben in standard German, as in das Kind hat nicht gefolgt (the child did not obey). When it means to go after someone, use sein.
In southern Germany, Austria and Switzerland you will hear ich bin gesessen, ich bin gestanden and ich bin gelegen. In the north these are ich habe gesessen and so on. Both are correct in their regions, but if you are unsure, use haben for these three.
The typical learner mistake is ich habe gewesen or es hat passiert. If you catch yourself translating English have, stop and check whether the verb is one of this group.
Verbs of being, staying and happening take sein in the perfect, even though nothing moves: ich bin gewesen, ich bin geblieben, es ist passiert, es ist geschehen, and also ist erschienen, ist gescheitert and ist gefolgt. Do not let English have push you towards haben, and remember the exception hat stattgefunden.