A Bar Mitsva - the event - is a traditon which is a few hundred years old.
When the boy turns 13 (according to the Jewish calendar) or a girl tuns 12, he/she are "off age". responsible for their doing, obliged to follow the mitsvot.
A boy who is 13 years old also counts for the minyan (I am not talking about Reform Judaism, this came up "only" a few hundred years ago) and as a symbol he reads a part (maftir) from the weekly Torah portion.
Also, traditionally, he has to lay t'fillin before that. As a consequence: As soon as he is 13 years old he demonstrates his status by laying t'fillin in the synagogue and on the following Shabbat he shows that he is also permitted to read from the Torah. This is called "Bar Mitsva". But halakhically it has no meaning. Of course a Bar Mitsva can also be held later (but not earlier). Veronika Pachtinger explained it very nicely (unfortunately I saw her post when I had almost finished mine).
In my certificate of the Brit Milla is even written when my Bar Mitsva is - the portion of the Tora. But I had it one week later, I do not know why and when I asked the cantor 20 years later he couldn't remember. My guess is that the portion I had to read was pretty lengthy, so he set the date a week later. The reading was not all that difficult.
On which day the boy turns 13 can easily be calculated by converting the Gregorian date of birth to the Jewish one.
I hope this explains the situation a bit.
David Seldner, Karlsruhe, Germany