Actually, we are not ready at this point in the season to request rain; it is a bit early, and the "olai regel," "the people who have gone up by foot," to come to Yerushalayim for the Holiday in fulfillment of the Biblical command, need to get home without getting drenched. Here, rather, we praise G-d as the source of the blessing of rain; in early December, we will actually make the official request. The formula for this praise is the phrase, "Mashiv HaRuach U'Morid HaGeshem," "the One Who makes the wind blow, and makes the rain descend."
When is this prayer (or phrase of praise) recited?
It is always done on Shmini Atzeret, but the exact time is a matter of custom. Some congregations begin saying it in the Silent Shemoneh Esrei (1) of Mussaf; others don't begin reciting it until the Silent Shemoneh Esrei of Minchah, the Afternoon Prayer.
Where is the prayer inserted?
The Shemoneh Esrei begins always with a Section of Praises of G-d, consisting of three blessings, which will be described here in extremely brief terms. The first, called "Avot," "forefathers," praises G-d as the G-d of our forefathers, as a great and awesome G-d, Who remembers their merit in our behalf. The next blessing extols G-d's Might as the Healer, the One Who gives life and takes it, and Who does not forget those who have died, and Who ultimately will bring about the resurrection of the dead. The third blessing in this Section describes G-d's Holiness, and describes Him as One Who is holy and Whose Name (i.e. His actions) are holy.
"Mashiv HaRuach U'Morid HaGeshem" is inserted into the blessing "describing" His indescribable Might, as the Source of life-giving rain.
You are eternally mighty, my Lord, the Resuscitator of the dead are You; abundantly able to save,
Who makes the wind blow and makes the rain descend;
Who sustains the living with kindness, resuscitates the dead with abundant mercy, supports the fallen, heals the sick, releases the confined, and maintains His faith to those asleep in the dust. Who is like You, 0 Master of mighty deeds, and who is comparable to You, 0 King Who causes death and restores life and makes salvation sprout! And You are faithful to resuscitate the dead. Blessed are You, HASHEM, Who resuscitates the dead.
The phrase is testimony to the centrality of water, H2O, to life and the marvelous delivery system, namely the wind, which transports it and arranges the climate for the entire planet.
Till when is this phrase included in the Prayers?
It is included until Pesach, when a prayer similar to "Geshem," meaning "Rain," namely "Tal," meaning "Dew," is recited. Dew is a sign of Spring, and one of the names of Pesach is the "Chag HaAviv," the "Festival of Spring." At that time, some communities have the custom of simply removing "Mashiv HaRuach;" others replace it with the phrase "Morid HaTal," the "One Who brings the Dew."
What are some laws about the recitation of "Mashiv HaRuach?" Specifically, what should one do if he or she forgets to recite that phrase?
There are several variations, depending on when one realizes that it was forgotten.
If one remembers before finishing the blessing of "Who resurrects the dead," say "Mashiv … " and finish the blessing.
If one remembers after finishing the above blessing, but before beginning the next one, which begins "You are holy," say "Mashiv … " and it is not necessary to start the Shemoneh Esrei again.
If one remembers after beginning "You are holy … ," return to the beginning of Shemoneh Esrei.
If one finishes the Shemoneh Esrei, and does not remember whether he or she said "Mashiv HaRuach … ," or not, if thirty days have passed since it was re-incorporated into the prayers, then one can assume that it was said (by force of habit, if nothing else); if not, one should assume that it was not said.
From the above details [and remember, "G-d is in the details"], we see that this phrase, which recognizes the greatness and the necessity of the delivery of moisture, is a key element in the Praise of G-d for, if the phrase is inadvertently omitted, the Shemoneh Esrei must be repeated. Also, that if a key phrase is omitted, especially one built into the very structure of the Shemoneh Esrei, and a new thought is begun, the entire Shemoneh Esrei must be repeated. And finally, that Halacha (2) recognizes the reality of repetition and habit formation in human behavior.