In 2019, Rosh Hashanah began yesterday at sunset and ends tomorrow evening, roughly, but not exactly (as expected in a lunar Jewish calendar) at the time of the autumnal equinox.
Rosh Hashanah is not mentioned in the Torah, Judaism’s founding religious text, and appears under different names in the Bible. Though the holiday was likely well established by the sixth century B.C.E, the phrase “Rosh Hashanah” shows up for the first time in the Mishna, a Jewish code of law compiled in 200 C.E [that is the first major written product of Rabbinic Judaism].
The Hebrew calendar begins with the month of Nisan, but Rosh Hashanah occurs at the start of Tishrei, when God is said to have created the world. For this reason, Rosh Hashanah can be seen as the birthday of the world rather than New Year’s in the secular sense; still, it is on Rosh Hashanah that the number of the civil year increases. The Mishna described three other “new years” in the Jewish calendar in addition to Rosh Hashanah. Nisan 1 was used to resume the cycle of months and measure the duration of kings’ reigns. Elul 1 resembled the start of the modern fiscal year and determined the tithing of animals for charity or sacrifice. Shevat 15 calculated the age of fruit-bearing trees and is now celebrated as the minor holiday of Tu B’Shevat.
According to tradition, God judges all creatures during the 10 Days of Awe between Rosh Hashanah and Yom Kippur, deciding whether they will live or die in the coming year. Jewish law teaches that God inscribes the names of the righteous in the “book of life” and condemns the wicked to death on Rosh Hashanah; people who fall between the two categories have until Yom Kippur to perform “teshuvah,” or repentance. As a result, observant Jews consider Rosh Hashanah and the days surrounding it a time for prayer, good deeds, reflecting on past mistakes and making amends with others.From here.
At the time these dates were established, the Jewish people subject to the people setting these dates lives in the Southern Levant with periods of "Babylonian exile" in Mesopotamia. The time period in question starts before Roman conquest and extends to a time period after it. The first written attestation of the date of Rosh Hashanah is after the diaspora begins (70 CE), but before either written historical accounts or genetic evidence suggests that there was mass migration of Jews to Europe.
In contrast, the Roman Empire's calendar that ultimately became the basis for the calendars of the Roman Catholic and Eastern Orthodox churches, which also served as the secular calendar of Europe after the fall of the western Roman Empire, started in the spring (which is why "December" the tenth month, is in the winter).
The Chinese lunar calendar also begins in the winter, typically in roughly January by the Roman calendar.
The Jewish lunisolar calendar developed multiple New Year's for different purposes. As Wikipedia explains:
The Jewish calendar has several distinct new years, used for different purposes. The use of multiple starting dates for a year is comparable to different starting dates for civil "calendar years", "tax or fiscal years", "academic years", and so on. The Mishnah (c. 200 CE) identifies four new-year dates:
The 1st of Nisan is the new year for kings and festivals; the 1st of Elul is the new year for the cattle tithe... the 1st of Tishri is the new year for years, of the years of release and jubilee years, for the planting and for vegetables; and the 1st of Shevat is the new year for trees—so the school of Shammai; and the school of Hillel say: On the 15th thereof.
Two of these dates are especially prominent:
* 1 Nisan is the ecclesiastical new year, i.e. the date from which months and festivals are counted. Thus Passover (which begins on 15 Nisan) is described in the Torah as falling "in the first month", while Rosh Hashana (which begins on 1 Tishrei) is described as falling "in the seventh month". Since Passover is required to be celebrated in the spring, it should fall around, and normally just after, the vernal (spring) equinox. If the twelfth full moon after the previous Passover is too early compared to the equinox, a 13th leap month is inserted near the end of the previous year before the new year is set to begin. According to normative Judaism, the verses in Exodus 12:1–2 require that the months be determined by a proper court with the necessary authority to sanctify the months. Hence the court, not the astronomy, has the final decision.
* Nowadays, the day most commonly referred to as the "New Year" is 1 Tishrei (Rosh Hashanah, lit. "head of the year"), even though Tishrei is the seventh month of the ecclesiastical year. 1 Tishrei is the civil new year, and the date on which the year number advances. Tishrei marks the end of one agricultural year and the beginning of another, and thus 1 Tishrei is considered the new year for most agriculture-related commandments, including Shmita, Yovel, Maaser Rishon, Maaser Sheni, and Maaser Ani.
Thus, to some extent, the Jewish calendar starts in the spring like the Roman calendar used to, and to some extent it starts in the fall, although autumn in the Southern Levant or Mesopotamia (like each of the other seasons in this region) bears little resemblance in climate to autumn in Europe or in northern North America.
The typical timeline of Rosh Hashanah is also a bit early for the last pre-winter harvest in a Mediterranean climate, which is usually more like late October or early November.
One plausible possibility is that he migration of the "most important" of the New Year's of the Jewish calendar from the spring start described in the Torah to the fall beginning that became normative at least in the Rabbinic era, if not earlier, reflects the transition of the Hebrews from being herders, as they were according to their own tradition immediately prior to settling in the Levant, to farmers embedded in a larger urban society. But, there don't seem to be definitive answers to this question.