I know I haven't posted in a long time, as I've been busy, but I have some time now, and this date happens only once every four years, so I won't let it pass without comment.
The truth is, calling the last day of February "leap day" is a relatively recent practice. Although we today count days from the start of the month, our Gregorian calendar was originally a slight modification of the Julian calendar, itself based by Julius Caesar on the earlier Roman calendar, which counted days backwards. Thus, the first day of each month was called the Calends or Kalends of that month, so the last day of February would be known as the day before (Latin Pridie) the Calends of March. (This is where we get the word "calendar".) The rest of the days in the last half of each month were likewise counted down to the Calends of the following month. The Romans included the Calends itself as the last day in their countdown.
Caesar did not insert the leap day at the end of February, but on the 24th day, which in Roman counting was the sixth day counting down to the Calends of March. Thus, in leap years two days were both counted as the sixth day before Calends, which we would now call the 24th and 25th days of the 29-day month. (Sometimes the 24th was considered the extra day and sometimes the 25th.)
Consequently, someone born on the last day of February in a leap year, i.e. February 29, would not have celebrated only once every four years, but rather every year on the last day of the month, which we call February 28 in common years. Since the extra day was counted as the same as the 24th, as though it were one 48-hour-long day, anyone born on the leap day, i.e. February 25, would celebrate their birthday in common years on February 24, the sixth day before the Calends of March. Likewise, the Roman Catholic Church recognized February 24 as the feast of St. Matthias in common years, but February 25 in leap years, up until as recently as 1970. Other feasts near the end of February were likewise shifted. I believe that this is still practiced by those using the Julian calendar, i.e. Orthodox churches, although in this century Julian calendar dates fall 13 days behind those in the Gregorian calendar.
22 23 24 25 26 27 28 1...
VIII VII VI V IV III II K
22 23 24 25 26 27 28 29 1...
VIII VII VI VI V IV III II K
So in actuality, the leap day occurred on Saturday, February 25, rather than today. There are a number of articles on the Internet today about leap years, so I'll link to How the Quest for the Perfect Calendar Accidentally Created February 30.