"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. (The Ashkenazic pronunciation is "bas")
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/Bat 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/Bat 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/Bat 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 celebrant is also generally required to make a speech, which traditionally begins with the phrase "Today I am a man/woman."
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/bat 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. There is not a Reform synagogue today that does 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/bat 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 ritual. Bar/Bat 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 Talmud 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/Bat mitzvah is simply the age when a person is held responsible for his actions and minimally qualified to marry.
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 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.