"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 become used to doing the obligations they will have to do as adults (and their fathers are responsible for preparing them for Torah observance). At the age of 13 (12 for girls), children become adults and are 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 Talmud, 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 was the celebrant's first aliyah for European Jews (Yemenite Jews always allowed even children of five or six to take an aliyah, which is the correct practice according to the Torah). During Shabbat services 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. The father typically recites a blessing thanking God for removing the burden of being responsible for the son's sins, which is based on two misunderstandings: there is not properly any such blessing according to the Torah, and the father who utters it will be punished for saying God's name in vain; moreover, fathers are not punished for their minor sons' sins and the sons are not punished for their sins as minors, either (under some circumstances, sons are punished additionally for their fathers' sins, but only when they continue the misbehavior of their fathers).
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, despite that there is no foundation for this according to the Torah.
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.
The Reform movement tried to do away with the Bar Mitzvah for a while, scorning the idea that a 13 year old child was an adult. They replaced it with a confirmation at the age of 16 or 18. However, due to the overwhelming popularity of the ceremonies, the Reform movement has revived the practice. We do not know of any Reform synagogues that do not encourage the practice of Bar and Bat Mitzvahs today. In some Conservative synagogues, however, the confirmation practice continues as a way to keep children involved in Jewish education for a few more years.
The age set for Bar Mitzvah is not an outdated notion based on the needs of an agricultural society, as some suggest. This criticism comes from a misunderstanding of the significance of the Bar Mitzvah. Bar Mitzvah is simply the age when a person is held responsible for his actions and minimally qualified to marry, because he is normally physically and mentally able to do so; in case the child has not developed normally, his resonsibility is delayed according to his actual development, and those who have permanent mental disability are forever exempt from the commandments and their punishment. The Torah sees normal young adults as adults, and there is really no need to be as pessimistic (and even as insulting) toward young people as modern Western society is.
If you compare this to secular law, you will find that it is not so very far from our modern notions of a child's maturity. In Anglo-American common law, a child of the age of 14 is old enough to assume many of the responsibilities of an adult, including minimal criminal liability. In many states, a fourteen year old can marry with parental consent. Children of any age are permitted to testify in court, and children over the age of 14 are permitted to have significant input into custody decisions in cases of divorce.
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