"Bar Mitzvah" literally means "son of the commandment." Under Jewish Law, children are not obligated to observe the commandments, although they are encouraged to do so as much as possible to learn the obligations they will have as adults. At the age of 13 (12 for girls), children become obligated to observe the commandments.
The Bar Mitzvah ceremony formally marks the assumption of that obligation, along with the corresponding right to take part in leading religious services, to count in a minyan (the minimum number of people needed to perform certain parts of religious services), to form binding contracts, to testify before religious courts and to marry.
A Jewish boy automatically becomes a Bar Mitzvah upon reaching the age of 13 years. No ceremony is needed to confer these rights and obligations. The popular Bar Mitzvah ceremony is not required, and does not fulfill any commandment. It is a relatively modern innovation, not mentioned in the Torah, and the elaborate ceremonies and receptions that are commonplace today were unheard of as recently as a century ago.
In its earliest and most basic form, a Bar Mitzvah is the celebrant's first Aliyah (going up) to the Torah. During Shabbat prayer services on a Saturday shortly after the child's 13th birthday, the celebrant is called up to the Torah to recite a blessing over the Parshat HaShavua (weekly reading).
Today, it is common practice for the Bar Mitzvah celebrant to do much more than just say the blessing. It is most common for the celebrant to learn the entire Haftarah portion, including its traditional chant, and recite that. In some congregations, the celebrant reads the entire weekly torah portion, or leads part of the service, or leads the congregation in certain important prayers.
Bar Mitzvah is not about being a full adult in every sense of the word, ready to marry, go out on your own, earn a living and raise children. The Torah makes this abundantly clear. In Pirkei Avot, it is said that while 13 is the proper age for fulfillment of the Commandments, 18 is the proper age for marriage and 20 is the proper age for earning a livelihood. Elsewhere in the Talmud, the proper age for marriage is said to be 16-24. Bar Mitzvah is simply the age when a person is held responsible for his actions and minimally qualified to marry.
Nevertheless, one should not celebrate a Bat Mitzvah in the same manner of a Bar Mitzvah. Rather, it should be celebrated in a modest manner in school, similar to a birthday. The parent's should buy her a nice present. Celebrate the Bat Mitzvah as part of a Mesibat Shabbat (Shabbat Party) or a Melaveh Malkah (Dinner after Shabbat).