A boy attains his religious maturity and becomes responsible to fulfill all Torah commandments at the age of 13. A girl becomes equally responsible one full year earlier, at the age of 12, corresponding to her earlier maturation.
At this age, young men and women gain the intellectual maturity to discern between right and wrong and modify their behavior accordingly.
Bar Mitzvah literally means "Son of Mitzvah," which describes the young man now responsible to do all mitzvot (commandments). Bat Mitzvah literally means "Daughter of Mitzvah," or the young woman now responsible to observe all of her mitzvot.
This occasion is cause for great celebration and gratitude to G-d, and hence the Bar or Bat Mitzvah ceremony. Note: One becomes Bar or Bat Mitzvah automatically on one's twelfth or thirteenth birthday, with or without a party!
For a boy, among his many new responsibilities, beginning from his Bar-Mitzvah day and onward he must wrap tefillin every day of the week besides Shabbat and certain holidays. Traditionally boys are called up to the Torah on the day of their Bar Mitzvah (or the Sabbath afterward), reflecting their new status as adult participants in the prayers. In some communities, the boy reads from the Torah and/or chants the Haftorah on the Shabbat after he becomes Bar Mitzvah.
The following links discuss the practical, philosophical and mystical implications of the Bar and Bat Mitzvah rite of passage.