What is the purpose of a school in a synagogue?

Yeshiva

A yeshiva is a school where Jewish men, from bar mitzvah until about age 18, or as long as age 22-- even longer, if they want to pursue a career as a rabbi-- go to study Talmud and Torah. This occupies their entire day, with the exceptions of prayers and meals. They usually stay in dormitories at the yeshiva, or board with nearby families, unless their own families happen to live nearby, but this is an exception. Some boys travel a great distince to go to yeshiva, some US boys go to Israel, for example.

A few modern Orthodox yeshivot offer education to girls, but they are exceptions. Traditional yeshivot are for men only. Men who have completed main studies may still go for a few hours a day after putting in a day working at a secular job. The best students may be asked to stay on as teachers.

Orthodox girls after bat mitzvat usually got to a bais Yaakov. Girls in these schools generally study the same kinds of core academics during one half of the day that regular public high school students study, and the other half studying Jewish law as it pertains to domestic life and childrearing, but that is a long and difficult course that covers a lot of ground, and also prepares girls, as adults to deal with the secular world as they might need to. The academic portion of study generally is rigorous enough to prepare girls for college programs.

Some post-bais Yaakov girls attend two-year programs at institutions called seminaries, where they learn more about the Torah and Talmudic basis behind a lot of what they were taught in bais Yaakov, and also learn more Jewish history. They may formally study Hebrew, but as some of the best seminaries are in Israel, they may learn Hebrew for practical reasons as well. (Some groups of Orthodox Jews do not feel it is right to use Hebrew as an everyday language, but this is not the majority of them, although you will not hear it as an everyday language just for the sake of doing so in communities outside of Israel.)

Private schools for children under bar and bat mitzvah age are usually referred to as "Jewish Day Schools," but up until about the 1960s, were called Talmud Torahs, and are still called this is Orthodox communities, sometimes.
It's called a yeshiva.