Bar Mitzvah" literally means "son of the commandment" and "Bat Mitzvah" means "daughter of commandment". Technically, the term refers to the child who is coming of age, but it is more commonly used to refer to the coming of age ceremony itself.
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 for boys and 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, and a girl upon reaching the age of 12 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. The ceremonies and receptions that are commonplace today were unheard of as recently as a century ago.
Many people mock the idea that a 12 or 13 year old child is an adult, claiming that it is an outdated notion based on the needs of an agricultural society. But 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 Talmud makes this abundantly clear. Bar mitzvah is simply the age when a person is held responsible for his actions and minimally qualified to marry.