Supposedly, the birthday upon which a person becomes the same age as the date of their birth. For example, turning 25 on the 25th of the month. This event is to be marked by the consumption of champagne... not always a feasible method of celebrating one's birthday, to be sure, especially if one was born quite early on in the month.
In the case of yours truly, this event will occur tomorrow... as I turn 27 on the 27th of June. «/shameless plug»