The Mishna in tractate Rosh Hashana (1, 1) says that there are four New Years. I know that there's the standard one for the holidays, one for kings, one for rain. What's the fourth? When are they and what do they represent?
The fourth new year is the year for determining which flock an animal belongs to for maaser calculations. Note that it is not the new year for rain, but the new year for trees.
1 Tishrei, 15 Shvat, 1 Nisan, 1 Elul. There is a machlokes in which the last one really occurs on 1 Tishrei. However, we still refer to the four New years because of the gemara on Rosh Hashana 7b (I will explain that at the end of this post).
The last new year, 1 Elul, is the New Year for the tithing of cattle. The tithe for cattle had to be made from cattle born in the same fiscal year, between 1 Elul one year and the next.
Rosh Hashana Mishna א (Talmud 2a) (translation from Art Scroll)
There are four new years. First of Nisan is the new year for (Jewish) kings and for the festival.
On the first of Elul is the new year for the maaser of animals. - However R' Eliezer and R' Shimon say: it is on the first of Tishrei.
Note that Rambam Bechorot 7:6 says that the halacha is like R' Eliezer and R' Shimon (hat tip to @DoubleAA)
On the first of Tishrei is the New Year for counting years, the shmittah and Yovel years, for the sapling and for the vegtables.
On the first of Shvat is the New Year for trees - according to Bais Shamai. Bais Hillel says on the fifteenth.
Art Scroll explains the discussion as to why we say four (and not five) new years. The fifth is because there are those who say that the New Year for holidays is Pesach itself (15 Nisan) which adds an extra New Year. This is part of the discussion as to how the year is counted for delaying the fulfillment of an oath. The Art Scroll has that discussion on Daf 7b2.
Even though Rebbi himself holds that there are five New Years, he stated that there are four because everyone agrees with at least four as the Gemara proceeds to explain (Rashi)
R' Meier holds that the first of Elul is the New Year for maaser, but he does not count the fifteenth of Nisan as a New year for festivals because (as was learned above, 4b) he holds that one is liable for delaying a vow after a single festival has passed (Rashi)
R' Shimon holds that the fifteenth of Nisan is a New Year for festivals. But he also counts only four New Years because he does not count the first of Elul as a a New year for the maaser of animals. Rather, he holds that that New Year falls on the first of Tishrei, which is already counted as one of the New Years for other matters (Rashi)
Rav Nachman Bar Yitzchak says that since the first and the fifteenth of Nisan are in the same month, then they are counted together to allow for four months with a New Year. (Rashi)
ארבעה ראשי שנים הם. (1) באחד בניסן ראש השנה למלכים. ולרגלים. (2) באחד באלול. ראש השנה למעשר בהמה. רבי אלעזר ורבי שמעון אומרים באחד בתשרי. (3) באחד בתשרי ראש השנה לשנים. ולשמיטין. וליובלות. לנטיעה. ולירקות. (4) באחד בשבט ראש השנה לאילן. כדברי בית שמאי. בית הלל אומרים בחמשה עשר בו:
The four new years are: (1) On the first of Nisan, the new year for the kings and for the festivals; (2) On the first of Elul, the new year for the tithing of animals; Rabbi Eliezer and Rabbi Shimon say, on the first of Tishrei. (3) On the first of Tishrei, the new year for years, for the Sabbatical years and for the Jubilee years and for the planting and for the vegetables. (4) On the first of Shevat, the new year for the trees according to the words of the House of Shammai; The House of Hillel says, on the fifteenth thereof.
(Sefaria translation; numbers added)