what is a 15th birthday celebration called
Hi and welcome to the forum.
This is called a "Quinceañera."
You asked about the celebration. I am used to the celebration being called "fiesta de quince años" or "fiesta de quinceañera". The birthday girl is "la quinceañera".
Its only celebrated for girls at their "quinceanera." This is when the girl turns into a woman. I've been to many quinces. I'm mexican so thats why i know so much.