"Bar Mitzvah" literally means "son of the commandment" "Bar" is "son" in Aramaic, which used to be the vernacular of the Jewish people "Mitzvah" is "commandment" in both Hebrew and Aramaic. "Bat" is daughter in Hebrew and Aramaic.
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, 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. In its earliest and most basic form, a Bar Mitzvah is the celebrant's first aliyah. During Shabbat 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 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. In modern times, the religious service is followed by a reception that is often as elaborate as a wedding reception.
In Orthodox and Chasidic practice, women are not permitted to participate in religious services in these ways, so a bat mitzvah, if celebrated at all, is usually little more than a party. In other movements of Judaism, the girls do exactly the same thing as the boys.
It is important to note that a bar mitzvah is not the goal of a Jewish education, nor is it a graduation ceremony marking the end of a person's Jewish education. We are obligated to study Torah throughout our lives. To emphasize this point, some rabbis require a bar mitzvah student to sign an agreement promising to continue Jewish education after the bar mitzvah.