Why do some verbs take avoir and some verbs take être in the perfect tense and how can I remember which ones?

Many methods will tell you that verbs of movement take être in the perfect tense but that's not very helpful, since many verbs of movement do not use “être”, such as “danser (to dance), sauter (to jump), courir (to run), marcher (to walk) etc…” which use “avoir”:

J’ai dansé, j’ai sauté, j’ai couru, j’ai marché…

A better explanation is that these verbs that take être are not just verbs of movement but intransitive verbs. Knowing their name however may not be of much help to you when trying to use them in your writing and speaking, which is why you can use the mnemonic method DR & MRS VANDERTRAMP

Devenir Revenir & Monter Rester Sortir

Venir Aller Naître Descendre Entrer Rentrer Tomber Retourner Arriver Mourir Partir et Passer par

Now, don’t forget that all reflexive verbs also use “être”. Je me suis lavée – (se laver) I washed myself.