While in most Western societies young people are recognized as adults from their 18th birthday onward, in Judaism this traditionally occurs when a girl reaches the age of 12 years + one day, and a boy the age of 13 years + one day. This change in status occurs in Western societies and in Judaism automatically, whether one celebrates / commemorates this occasion or not.
In Judaism, this new status indicates that the Bat or Bar Mitzvah person is now obligated to observe Mitzvot (Religious Commandments) as all other Jewish adults are obligated. With these obligations, come certain privileges, such as being able to lead services and read from the Torah.
Originally, in the Middle Ages when Judaism began to develop ways of celebrating this transition, the Bar Mitzvah boy was only expected to read/recite the blessings before and after a section of Torah was read. As this tradition developed, the Bar Mitzvah boy was given more and more responsibility for leading the entire service according to his capabilities. Fortunately, we currently live in an egalitarian era so for nearly a hundred years now Bat Mitzvah girls are granted the exact same privileges as Bar Mitzvah boys with regard to Jewish ceremonial practices. Differences in practice for boys and for girls, vary, of course, from congregation to congregation.