The age of thirteen is first mentioned (although loosely) in the Orach Chayim which says that on a boy's thirteenth birthday his father should make a festive meal for the boy. Avot 5 states that at the age of thirteen a boy may fulfil the mitzvot (commandments). Thus, a boy (or girl for a bat mitzvah) is seen as an equal to any adult in the ways of Torah and the Law once they are bar mitzvah. This is also the age of responsibility, where the 13 year old becomes responsible for their own actions.

There are two commandments which may be followed after bar mitzvah which were previously disallowed - that of laying Tefillin and that of reading from the Torah. The boy is accepted into the community, and may hence be counted in the minyan, the ten men or, in the case of a girl, separately, women, who are required to be present for certain prayers to be recited.
Women do not have Bar Mitzvah's, they traditionally have Bat Chayils, the nominal equal of a Bar Mitzvah, only occuring a year earlier, at the age of 12, due to the fact that girls mature faster than boys.
The lavish parties and dinners for the modern bar mitzvah/bat chayil are, in fact, contrary to Jewish thought, where the father lays on a modest meal and then the boy leads a discussion on any halachic subject. The lavishness is a relatively new addition, as indeed is the ritual of bat Chayil, which was introduced in the ways of equality only a generation ago.

Bar and bat mitzvah are not one-off events. The actual ritual does take place on one day, but afterwards the person is always bar mitzvah. They are always a part of the community, always able to read from the Law, and always able to perform the mitzvot. Bar mitzvah is a state of mind as well as an excuse for a party!