Welcome to Maseches Rosh Hashanah. The tractate deals both with the laws of Rosh Hashanah and with the laws of Rosh Chodesh - determining when the new month begins and on which day it is fixed. The need for this arises from the fact that the moon's orbit around the earth takes twenty nine and a half days and a little more, and therefore a month can be counted as twenty nine days or as thirty days. This determination is one of the subjects we will deal with in the first chapter of the tractate.
The first Mishnah, in chapter one, deals with different kinds of Rosh Hashanah - and not only with the Rosh Hashanah familiar to us, which falls on the first of Tishrei. The words of the Mishnah: "Arba'ah rashei shanim hem" - there are four different dates, each one of them a new year for a different matter.
"B'echad b'Nisan Rosh Hashanah la'melachim v'la'regalim":
For kings - this refers to the count of the years of a king's reign, according to which documents were dated. A king who ascended the throne in the middle of the year is considered, once the first of Nisan has passed, to have entered the second year of his reign: the year in which he became king is the first year, and on the following first of Nisan a second year begins.
The practical halachic significance: since documents were dated according to the years of the king, a mistaken date could turn the document into a predated one, presenting the loan as though it had been made earlier than it actually was. The holder of a promissory note collects from property that the borrower sold, but only from the time the loan was made and onward. A document dated earlier than the truth would allow the lender to collect property that had been sold to a third party before that time - a collection to which he was not entitled.
For the festivals - here the matter relates to the prohibition of "bal te'acher." The Torah establishes that one may not delay bringing a korban or a donation that he vowed to the Beis HaMikdash beyond a certain time. Our Mishnah follows the opinion of Rabbi Shimon, who holds that a person does not transgress the prohibition until the three festivals - Pesach, Shavuos and Sukkos - have passed, and specifically in that order, with Pesach first. Therefore, for one who vows after Pesach, we count Shavuos and Sukkos, and afterward Pesach, Shavuos and Sukkos. From this perspective the first of Nisan represents Pesach, and serves as a kind of new year for this matter.
"B'echad b'Elul Rosh Hashanah l'ma'aser beheimah":
Ma'aser beheimah is the obligation to separate one out of every ten of the animals born in the flock during that year. Animals born before the first of Elul are not combined with those born after it, since they belong to two separate years, and the tithe is separated independently from the group before this new year and from the group after it. "V'elu d'Rabbi Shimon omrim" - in their opinion, the new year for ma'aser beheimah falls on the first of Tishrei.
"B'echad b'Tishrei":
"L'shanim" - here too the reference is to the count of the years of kings and its significance for documents, as explained above. The difference is that the first of Nisan applies to the kings of Israel, while the first of Tishrei applies to the kings of the nations of the world.
"L'shemittin v'la'yovlos" - this is the date on which the shemittah year begins, and likewise the yovel year every fiftieth year. For some of the laws, the yovel actually begins on Yom Kippur of that year.
"La'neti'ah" - one who plants a new tree may not eat or derive benefit from its fruit during the first three years, and in the fourth year the fruit must be brought up to Yerushalayim. For one who plants in the middle of the year, once the Rosh Hashanah of the first of Tishrei has passed, this is considered the second year of the tree.
"La'yerakos" - regarding the tithing of vegetables, the separation of a tenth of the produce. One does not separate terumah and ma'aser from one year's produce on behalf of another year's; rather, the tithe is taken separately from each and every year, and the dividing line between the years is the first of Tishrei.
"B'echad b'Shevat Rosh Hashanah la'ilan":
This is the new year for trees "k'divrei Beis Shammai", while "Beis Hillel omrim ba'chamishah asar bo" - this is the day known as Tu BiShvat. The dividing line established here relates to the fruit that begins to grow after the blossom falls off, called in halachic terminology 'chanatah': if the chanatah occurred before the first of Shevat according to Beis Shammai, or before the fifteenth of Shevat according to Beis Hillel, this determines the year to which the fruit belongs regarding the separation of the tithes of fruit.
In summary: in this Mishnah we learned the four new years: the first of Nisan - for kings (for counting the years of their reign and for dating documents) and for the festivals (regarding "bal te'acher"); the first of Elul - for ma'aser beheimah, and according to Rabbi Elazar and Rabbi Shimon it is the first of Tishrei; the first of Tishrei - for the years of the kings of the nations of the world, for shemittah and yovel, for planting and for vegetables; and the first of Shevat according to Beis Shammai, or the fifteenth of that month according to Beis Hillel - the new year for the tree regarding the chanatah of the fruit and the separation of its tithes.