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1. The Mishnah lists four different days that serve as the "new year" for various matters.
2. The first of Tishrei is the new year for five (or six) things.
3. The Mishnah does not mention the fact that Tishrei is the Rosh Hashanah for mankind (according to some Amora'im, see 8a).
4. There is a dispute about when the new year for taking Ma'aser from trees begins.
5. The Gemara explains how the first of Nisan is considered a new year for kings.
A BIT MORE
1. For example, the tithing of animals is done with animals that were all born during a particular year. The Tana Kama says that the year is counted from the first of Elul to the first of the next Elul. Rebbi Elazar and Rebbi Shimon say that it is counted from the first of Tishrei to the first of the following Tishrei.
2. They are: Years (see 8a), Shemitah, Yovel, Orlah, and Ma'aser of vegetables (and Ma'aser Behemah according to Rebbi Elazar and Rebbi Shimon).
3. The Amora'im who say that "years" does not refer to Rosh Hashanah of mankind understand that since this will be said in the Mishnah later (16a), it therefore is not said here.
4. Beis Shamai: It starts from the first of Shevat. Beis Hillel: It starts from the fifteenth of Shevat (Tu b'Shevat).
5. When they used to write the date in legal documents based on years of the current king's reign, the years of his reign were counted from the first of Nisan to the first of Nisan. Accordingly, if he came to power at the end of Adar, he would be considered as being in the second year of his kingdom during the next month, Nisan (and possibly even after one day if he became king at the very end of Adar).
Next Daf Index to Revach for Maseches Rosh Hashanah