Leap days date back to Julius Caesar, of course, but for our Gregorian calendar we can thank Pope Gregory XIII, as the Catholic Encyclopedia explains:
The necessity of a reform was continually urged, especially by Church authorities, who felt the need in connexion with the ecclesiastical calendar. It was accordingly strongly pressed upon the attention of the pope by the councils of Constance, Basle, Lateran (A.D. 1511), and finally by Trent, in its last session (A.D. 1563).
Nineteen years later the work was accomplished by Pope Gregory XIII (from whom the Gregorian reform takes its name) with the aid chiefly of Lilius, Clavius, and Chacon or Chaconius. There were two main objects to be attained: first, the error of ten days, already mentioned, which had crept in, had to be got rid of; second, its recurrence had to be prevented for the future. The first was attained by the omission from the calendar of the ten superfluous days, so as to bring things back to their proper position. To obviate the recurrence of the same convenience, it was decided to omit three leap years in every four centuries, and thus eliminate the three superfluous days, which, as we have seen, would be introduced in that period under the Julian system. To effect this, only those Centurial years were retained as leap years the first two figures of which are exact multiples of 4--as 1600, 2000, 2400--other centurial years 1700, 1800, 1900, 2100, etc.--becoming common years of 365 days each.
Meanwhile, some Orthodox Churches celebrate most of the great feasts of the liturgical year according to the "New Calendar," while most have retained the Original Julian.
The Orthodox (some Orthodox?) refer to the calendar adopted in Catholic countries in 1582 as the "New Style" Julian Calendar, while in 1923 a conference of Orthodox bishops adopted the "Revised" Julian Calendar, which will coincide with the "New Style" until about the year 2200. In the 19th century, bishops of the Russian Orthodox Church seriously considered adopted of the "New Style."
Calendrical reform is a sensitive issue, I see; from the outside, I'm inclined to look at the cleavages rather sociologically, with the adoption or rejection of a Western calendar serving as a stand-in for all sorts of other issues. Really I'm just poking around into arcana I'm curious about, much akin to how I've quizzed Jewish friends about various observances of theirs.
All Orthodox Churches (except the tiny Orthodox Church of Finland--who knew!) still use the Old Calendar for the date of Easter. Sometimes Easter for the Orthodox (and for Ukrainian Catholics among others) falls as much as five weeks later than the date celebrated by Protestants and Roman Rite Catholics, as indeed happens this year, when it will be April 27 (in the Gregorian calendar).
So now you know. Happy leap day, everyone.