A Bar/Bat Mitzvah traditionally takes place when a boy reaches 13 and a girl 12. Once the child reaches this age they may participate in a minyan, serve as witnesses to Jewish legal documents and they are obligated to the mitzvot that God gave us at Sinai. Whether or not this moment in time is marked with a ceremony, the Jewish child becomes an adult in the eyes of the community when they they turn 13 (for boys) and 12 (for girls).
According to various rabbinic authorities, this moment in time marks a new stage of intellectual, physical and moral development. The term bar/bat mitzvah essentially means that they are now "obligated to perform the mitzvot (commandments)." This transitional moment reflects the new Jewish adult's newfound capabilities and responsibilities.
I like to describe the Bar/Bat Mitzvah ceremony as walking through the doorway to Jewish adulthood. More specifically, the Bar/Bat Mitzvah is the beginning of walking down a hallway that leads to religious and communal responsibilities as a Jewish adult.
The process of celebrating a Bar/Bat Mitzvah often requires the child to learn how to read Torah, lead the prayers in a service and give a derasha--a brief sermon. These surmountable challenges require the child to work hard and attain a level of Jewish competence. It is the hope of the Jewish community that this simcha marks one's entry into a world of commitment to a Jewish way of life, filled with the observance of mitzvot, customs and traditions of our people.