Gemara

2a - Mishnah: We do not request rain (Ten Tal u'Matar in Birkas ha'Shanim) until near the time (that it normally comes);

R. Yehudah says, the Shali'ach Tzibur for Musaf of Shemini Atzeres mentions rain (Meshiv ha'Ru'ach u'Morid ha'Geshem in Techiyas ha'Mesim). The Shali'ach Tzibur for Shacharis does not mention rain.

10a (Mishnah): We request rain starting on the third of Cheshvan;

R. Gamliel says, we start on the seventh (of Cheshvan), 15 days after (the end of) Sukos, in order that the slowest travelers in Yisrael will reach the Pras river.

Gemara - R. Elazar: The Halachah follows R. Gamliel.

(Beraisa - Chananyah): In Golah (Bavel) we begin 60 days after the Tekufah (of Tishrei, the fall equinox).

Shmuel: The Halachah follows Chananyah.

Question: But Shmuel was asked when we mention Ten Tal u'Matar, and he said 'when people bring wood to the storehouse (for rain will come soon and it will not be available in the forest)'!

Answer: Perhaps this coincides with 60 days after the Tekufah.

Question: Do we request on the 60th day itself?

Answer #1 (Rav): Day 60 is like after 60 (we request);

Answer #2 (Shmuel): Day 60 is like before 60.

Rav Papa: The Halachah is, day 60 is like after 60.

4b - Rav Asi citing R. Yochanan: The Halachah follows R. Yehudah (we mention rain in Musaf of Shemini Atzeres, but not in Shacharis).

Question (R. Zeira): But R. Elazar said that the Halachah follows R. Gamliel (we do not request until the seventh of Cheshvan)!

Answer #1 (Rav Asi): R. Yochanan argues with R. Elazar.

Answer #2: R. Yochanan discusses mentioning rain. He agrees that we do not request until later.

Question: But R. Yochanan said that when we request, we also mention!

Answer: He means that when we cease to request, we cease to mention.

Objection: R. Yochanan said that when we begin to mention, we begin to request; when we cease to request, we cease to mention!

Answer #3: Rather, R. Yochanan discusses Eretz Yisrael; R. Elazar discusses Bavel.

Question: In Bavel we delay requesting because we have produce in the field at this time (and rain would be bad for it). In Eretz Yisrael they should also delay requesting, because Olei Regalim need to return home!

Answer: R. Yochanan discusses after the Churban (there are no Olei Regalim).

Answer #4: R. Yochanan and R. Elazar both discuss Eretz Yisrael. R. Elazar discusses when the Mikdash stands, and R. Yochanan discusses after the Churban.

Rishonim

Rif and Rosh (1:4): The Halachah follows R. Gamliel. The Halachah follows Chananyah. Day 60 is like after 60.

Ran (2a DH Masnisin): Eretz Yisrael is 400 Parsa'os by 400 Parsa'os, and a person walks only 10 Parsa'os (about 40 kilometers) in a day. Even if Yerushalayim was in the middle of Eretz Yisrael, one could not reach the border in less than 20 days. All the more so more time is needed, since Yerushalayim is close to Yericho near the Yarden, a border of Eretz Yisrael! Nevertheless, since rain is needed, Chachamim did not want to delay the request longer than the seventh, which is the time for the middle seasonal rainfall according to R. Meir. It is possible for people to travel quicker, e.g. if they also travel at night.

Ran (DH v'Ika): R. Elazar rules like R. Gamliel to teach that since in Eretz Yisrael they do not request until the seventh, any place where rain is harmful during this time does not request until the seventh. Any place where rain is needed and desired, like Eretz Yisrael, requests immediately after Sukos. If rain is needed but it would harm produce in the fields, we do not request until the seventh of Cheshvan. Places like Bavel that do not need rain do not request until 60 days after the Tekufah. The Rif and Rambam do not mention produce in the fields; they rule like R. Gamliel and Chananyah. They hold that even today in Eretz Yisrael we wait until the seventh. Even today, people gather to go to Yerushalayim on the Regel; for their sake we delay requesting rain.

Rambam (Hilchos Tefilah 2:16): From the seventh day of Cheshvan we request rain in Birkas ha'Shanim (and continue to request) as long as we mention rain. This refers to Eretz Yisrael. In Bavel, Surya, Mitzrayim and places close to them and similar to them we request rain on the 60th day after the Tekufah.

Rosh (ibid.): Avi ha'Ezri brings from the Yerushalmi that we count 60 days of 24 hours from the Tekufah. Sometimes we begin to request during the day, sometimes at night. The custom is unlike this.

Hagahos Maimoniyos (2:10): The Bavli does not mention counting 24 hour days, so we count from the day of the Tekufah.

Question (Rosh ibid.): Why do we follow the custom of Bavel? Granted, we follow the Talmud Bavli, but this depends on climate! Bavel was a lagoon and did not need rain, but why should lands that need rain in Cheshvan wait until 60 days after the Tekufah?! It is known that if rain did not come before this, the seeds would perish. Why don't we follow our Mishnah? In Province they requested rain in Cheshvan; I was very pleased.

Korban Nesan'el (4): Nevertheless, the Halachah does not follow this. If someone requested rain before 60 days he must go back.

Poskim

Shulchan Aruch (OC 117:1): In winter one must say Ten Tal u'Matar in Birkas ha'Shanim. In Chutz la'Aretz we begin to request in Ma'ariv on the 60th day after Tekufas Tishrei.

Rema: The day of the Tekufah is included in the 60.

Source (Gra DH v'Yom): This is like R. Yosi, who says that the day of the Tekufah begins the new Tekufah (Sanhedrin 13a, regarding whether or not we must Me'aber the year to make Sukos fall in the new Tekufah).

Beis Yosef (DH v'Chosav ha'Rav): The Avudraham says that the 60th day is the 22nd of November, or the 23rd if the coming February has 29 days.

Note: Rishonim bring a calculation of Tekufos of Rav Ada (it is not found in our Bavli or Yerushalmi - ha'Shomayim Mesaprim p.271). It is based on the 19 year cycle of leap years. Shmuel (Eruvin 56a) established that we count 91 days, seven and a half hours between successive Tekufos; a year of (four) Tekufos is 365 and a quarter days. Really, a year of Tekufos is slightly less, about 365.2425 days. (In 100 years Shmuel's calculation adds 25 days; really, there are 24.25 extra days. The Gregorian calendar averages 24 and a quarter leap years (on average) every century to compensate.) Shmuel knew the precise calculation, but enacted to use an approximation, so that even Yisraelim in Galus far from Chachamim can easily calculate when to request rain. He knew that Moshi'ach will come before the year 6000, so the total error of the calculation will not exceed 15 days (Sefer ha'Ibur, Igros Moshe OC 4:17).

Nowadays in Chutz la'Aretz we begin on December 4th (or 5th, before a leap year), 12 days later than the Avudraham says. The Gregorian calendar used to make every fourth year a leap year. In 5343/1583 they added 10 days, and enacted that years divisible by 100 but not by 400 (e.g. 1700) will not be leap years (Ta'amei ha'Minhagim 827 (Ha'arah) DH v'Zos (p.359); Hagahos Tur ha'Shalem). Shmuel's Tekufah falls a day later every year before a year divisible by four. Usually the next Gregorian year is a leap year, and it likewise adds a day. Each time the next year is not a leap year, e.g. 1700, 1800, 1900..., Shmuel's Tekufah permanently gets another day later in the Gregorian calendar. We should begin 13 days later than the Avudraham! Perhaps the Avudraham counted 60 days excluding the day of the Tekufah.

Kaf ha'Chayim (3): We follow Shmuel's calculation of Tekufos regarding this law, but we follow Rav Ada's calculation regarding Mo'adim (making Ibur Shanah so that Sukos and Pesach will fall in the proper Tekufos).

Beis Yosef (DH u'Mah she'Chosav mi'Kol): In a Teshuvah, the Rosh says that we follow the Bavli regarding Isur v'Heter, but we need not follow it regarding something that is based on the needs of the time that cannot entail an Isur Torah.

Magen Avraham (114:1): We wait to begin mentioning rain until Musaf (of Shemini Atzeres) in order that everyone will begin together. We are not concerned for this regarding request, and we start at night without an announcement, because in any case people in different climates start to request at different times.

Mishnah Berurah (3): After Kaddish before Shemoneh Esre, the overseer of the Beis ha'Keneses should announce 'Tal u'Matar'. Even if he did not do so, everyone must say it.

Kaf ha'Chayim (5): This should be announced at all Tefilos of the previous day, for its time is not fixed.

Shulchan Aruch (ibid.): In Eretz Yisrael we begin on the night of the seventh of Cheshvan.

Source (Gra DH uv'Eretz): The Rif and Rambam hold that the Halachah does not follow the Sugya on Daf 4b because it is not brought in the proper place (the Mishnah on 10a). Also, on 10a the Gemara simply says 'the Halachah follows R. Gamliel'. This connotes even nowadays.

Mishnah Berurah (5): Some say that if a resident of Eretz Yisrael is in Chutz la'Aretz or vice-versa, he begins at the same time as his origin only if he intends to return to his home within the year. If not, he begins like the place he is in. Others say that he (always) begins like the place he is in. However, it is possible that the source for the latter opinion discusses one who does not intend to return.

Note: Nowadays some suggest that one who will return within the year request in Shome'a Tefilah.

Kaf ha'Chayim (12): According to the Radvaz, if a Ben Eretz Yisrael in Chutz la'Aretz who intends to return home began requesting, and he (forgot and) became Shali'ach Tzibur before 60 days, he should request in his silent Shemoneh Esre but not in the repetition.

Kaf ha'Chayim (13): If a Ben Eretz Yisrael started requesting after the seventh of Cheshvan and went to Chutz la'Aretz before 60 days, he should continue to request there.