Welcome to Maseches Taanis. The tractate deals with the laws of fast days, and much of its discussion revolves around situations in which the rains are withheld - an issue that remains relevant in Eretz Yisrael to this very day. That process is described throughout most of the tractate.
The opening of the tractate, however, deals specifically with the prayers for rain, and within them two separate matters:
Hazkarah (mention) - saying "mashiv haruach umorid hageshem," which is a mention of the mighty deeds of Heaven: the might of the Holy One, blessed is He, as it is revealed in rain.
She'eilah (request) - the request for rain in the words "v'sein tal umatar" later in Shemoneh Esrei, which begins at a later date.
The words of the Mishnah:
"Mei'eimasai mazkirin gevuros geshamim?" - from when do we begin to mention the mighty deeds of Heaven, namely, saying "mashiv haruach umorid hageshem"?
The view of Rabbi Eliezer:
Rabbi Eliezer says: "Miyom tov harishon shel chag" - from the first day of Sukkos. His view is derived from the taking of the lulav: the lulav too comes to plead for rain and for water, concerning which we are judged on Sukkos. And just as the taking of the lulav begins on the first day of Sukkos, so too we mention the mighty deeds of Heaven by saying "mashiv haruach" already in the prayers of the first day of the festival.
The view of Rabbi Yehoshua:
Rabbi Yehoshua says: "Miyom tov acharon shel chag" - from the last day of Sukkos, which is Shemini Atzeres, and this is the practice in halachah.
The give and take between them:
"Amar lo: hoil v'ein hageshamim ela siman kelalah bechag" - rain during Sukkos is a sign of a curse, since it prevents us from fulfilling the mitzvah of dwelling in the sukkah, and if so, "lamah mazkir?" - why should one mention it?
Rabbi Eliezer answered him: "Af ani lo amarti lish'ol, ela lehazkir" - I did not say that one should request rain by saying "v'sein tal umatar" on Sukkos, but rather "lehazkir mashiv haruach umorid hageshem be'onaso" - merely to mention that the rains should fall in their proper season, after Sukkos.
Rabbi Yehoshua answered him: "Im kein l'olam yehei mazkir" - if that is so, one should mention throughout the entire year that the Holy One, blessed is He, brings down rain in its proper time, which is after Sukkos.
And indeed Rabbi Eliezer holds that, as far as the matter itself is concerned, one could have mentioned "mashiv haruach" all year round. It is only that one must begin it from the start of Sukkos, because it is fitting to place matters of praise and appeasement before the request. Therefore we first mention the might of the Holy One, blessed is He, in the rains, and after Sukkos we begin to request rain.
In summary: in this Mishnah we learned the distinction between mentioning the mighty deeds of rain ("mashiv haruach umorid hageshem") and requesting rain ("v'sein tal umatar"), and the dispute between Rabbi Eliezer and Rabbi Yehoshua as to when the mention begins: according to Rabbi Eliezer, from the first day of the festival, like the taking of the lulav which comes to plead for water, and according to Rabbi Yehoshua - and this is the halachah in practice - from the last day of the festival, since rains on Sukkos are a sign of a curse. And we learned the principle that emerges from Rabbi Eliezer's words: praise and appeasement come before the request.