Old vs. New Style means pre- vs. post-Gregorian Calendar reform.
From the OED's "IV. 27. a." definition for "style" (cf. esp. the last ¶):
A mode of expressing dates. Chiefly, Either of the two methods of dating that have been current in the Christian world since the introduction of the Gregorian calendar in 1582: viz., the New Style (abbreviated N.S.), which is the result of the Gregorian reform, and the Old Style (O.S.) which follows the unreformed calendar. The New Style is occasionally called the Roman Style, and the Old Style the English Style. In historical dates earlier than 1582, however, Roman Style, as used by modern writers, means only that the year mentioned is to be understood as beginning on 1 Jan.
The Julian calendar was based on the assumption that the tropical year consisted of 365¼ days. In order that the average calendar year should have this length, it was provided that the normal year should contain 365 days, but every fourth year 366 days. Down to A.D. 1582 the Julian calendar continued to be used by all Christian nations. In calendars and almanacs, the year began on 1 Jan. (like the Roman consular year); but for ordinary purposes the time of beginning the year was different in different places; in England, after some fluctuations, the beginning of the legal year was fixed for 25 March [Feast of the Annunciation]. After the adoption of the Christian era, the leap years were those whose number A.D. (reckoned from 1 Jan.) was divisible by 4.
The Julian estimate of 365¼ days for the length of the tropical year was too great by about 11 minutes, an error which amounts to one day in about 128 years. Hence in 1581 the date of 21 March for the vernal equinox, assumed since the early 4th c. in the rule for computing Easter, was 10 days too late. To remedy this inconvenience, and to prevent its recurrence, Pope Gregory XIII, acting on the advice of the Jesuit Clavius and other eminent astronomers, ordained that in A.D. 1582 the day after 4 Oct. should be reckoned as 15 Oct., and that in future the years which had a number ending in two cyphers should not be leap years unless the number were divisible by 400. The Julian date of 1 Jan. for the beginning of the year was retained. The difference between the old and new calendars continued to be 10 days until 1700 (the first disputed leap-year), when it became 11 days; in 1800 it became 12 days, and in 1900 13 days, from which there will be no further increase till 2100.
The Gregorian calendar (so called from the name of the Pope) was speedily adopted in all Roman Catholic countries, while the other nations of Europe adhered to their traditional reckoning. In the 17th and 18th centuries, it was often found necessary to state whether a date was according to Old or New Style, or to give both datings. As the nations which accepted the reform usually began the year on 1 Jan., not, as in England, on 25 Mar., there was for the March quarter (in addition to the other difference) a discrepancy in the number of the year between the Old Style and New Style dates.
In England and Scotland the Gregorian calendar was established by the Act 24 Geo. II. c. 23 (1751), which provided that the year 1752 and all future years should begin on 1 Jan. instead of 25 Mar. (in Scotland this rule had been adopted in 1600), that the day after 2 Sep. 1752 should be reckoned the 14 Sep., and that the reformed rule for leap year should in future be followed. Ireland followed in 1788. The use of New Style is now universal throughout the Christian world with the exception of certain countries of the Greek Church; in Russia it was officially adopted by the revolutionary government in 1918.
The use of stilus for ‘mode of dating’ was current in medieval Latin, as a specific application of the sense ‘usage’ (cf. 19 above[: A "style" (stilus) is "A method or custom of performing actions or functions, esp. one sanctioned by usage or law."]). In France the expression New Style (nouveau style) had been current before the time of the Gregorian reform, with reference to the change in the beginning of the year from Easter to 1 Jan., which took place in that country in 1563.
Building upon this definition of stylus, see
Herald of Landsperg reckoned the year from Christmas day. This was the “Christmas style” or stylus nativitatis, sometimes also called stylus curiae Romanae since it was often used by the Papal chancellery. In the civil calendar it was more common to use the stylus communis which placed new year's day on 1 January; since this was the feast of the circumcision of Jesus this system was also called stylus circumcisionis. In another system the year began in March as in the pre-Julian Roman and some old Germanic calendars. Still another stylus annunciationis or stylus incarnationis placed new year's day on 25 March which was both the feast of the Annunciation and the old Roman vernal equinox. This custom prevailed in England until the Gregorian Calendar was introduced in 1752. Finally one also had a movable year beginning at Easter, either on Good Friday or on the following day after the benedictio fontis. This motley of styles is one of the reasons of the intricacies of Medieval chronology.