Ever wonder why September is, etymologically speaking, the "seventh month," October the eighth, and so on?
This is a consequence of the Julian Calendar, which held sway in Europe from Roman times until 1582, when Pope Gregory XIII imposed the present system. Prior to 1582, March 25 was celebrated as New Year's Day, and March was considered the first month of the calendar year (even though only seven days of March fell within the new year). Worse, the Julian calendar required leap years every four years without exception, leading to a "drift" in dates. Equinoxes and solstices were still tied to their celestial hitching posts, but the dates on which they occurred drifted as much as ten days.
Gregory's new calendar was a sensible solution to these problems, and therefore was ignored by the British. In England and its colonies, the Julian remained the de jure calendar, even as the Gregorian became the de facto calendar. The most troublesome problem for those recognizing both calendars was designating dates from January 1 through March 24. On the Julian calendar, these were the last days of the old year, while on the Gregorian calendar they were the first days of the new year.
An example from my own family demonstrates the confusion that could arise from the Julian calendar. Jonathan Coolidge, son of John and Mary (Ravens) Coolidge of Watertown, Massachusetts, was born (according to town records) on the tenth day of the first month of 1645. A glance at adjacent records shows that the clerk was using the Julian calendar, whose first month was March. But in the Julian system, the first day of the first month is March 25. Undoubtedly, Jonathan was born on the tenth of March, but the year of his birth was actually 1644 (by the Julian calendar). The clerk's record is both correct—Jonathan was born in the first month of 1645, and on the tenth day of that month—and incorrect—Jonathan was born on the tenth day of March, which lay in 1644.
Such confusion led to a compromise: "double dating," or "split year dating." For dates on the Gregorian calendar prior to March 25, both years would be given. For example, "March 4, 1643/4" was both "March 4, 1643" (Julian) and "March 4, 1644" (Gregorian). This convention was widely used until the adoption of the Gregorian calendar in 1752.
For genealogists, it is important both to understand and to respect the double-dating system. When transcribing a record, one should always write the date as given—either in Old Style (Julian) or in New Style (Gregorian). For dates prior to 1752 written without double dating (e.g. "March 4, 1733"), one should not assume the use of either calendar without evidence. If one can reach a conclusion, the conclusion should be bracketed to prevent confusion (e.g. "March 4, 1733[/4]"). If evidence is less than conclusive, this should be indicated as well (e.g. "March 4, 1733[/4?]").