Hoshanah Rabbah is a very holy and special day. It means the “great salvation” and is celebrated on the seventh day of Sukkos. The judgment which was written on Rosh Hashanah and sealed on Yom Kippur is handed down on this day.
The Yalkut Shimoni explains that it is on the 21st of the month because Hashem’s Name of “I shall be” is equivalent to this in Gematria.
The davening for Shacharis is like Shabbos morning and the baal tefillah wears a kittel. We circle the bimah seven times on Hoshana Rabbah; during the regular days of Sukkos we circle the bimah only once. We also take a bundle of aravos and hit them on the ground until the leaves fall off.
There is also a custom to stay up all night on Hoshana Rabbah and study Torah. Chassidim have the custom to read from Sefer Dvarim on the night of Hoshana Rabbah. This is done without a bracha. There is a debate between Rav Elyashiv and Dayan Yitzchok Weiss (Minchas Yitzchak VIII #84) as to whether someone from Minhag Ashkenaz may participate in this unique nighttime Torah reading.
Shemini Atzeres is the culmination of Elul, the Yamim Nora’im, and Sukkos, times and holidays where we have developed an intense closeness with Hashem. But Shemini Atzeres is actually a new and independent Yom Tov, where Hashem says to us, “Stop. Don’t go. Stay with Me one last day.” This is the explanation of the word “Atzeres — stop.”
The number eight, according to the Maharal, alludes to the fact that this Yom Tov lies beyond the natural. There are seven days of the natural order — the eighth is above and beyond the natural. Shemini Atzeres lies beyond the natural order of the world. On Shemini Atzeres there was a gilui Shechinah, a revelation of the Divine Presence, so great that it filled the entire Mishkan (see Shabbos 87b). It was a light far beyond the natural order of this world.
The new Yom Tov necessitates its own Shehecheyanu. The berachah of Shehecheyanu should not just be said perfunctorily, Heaven forbid. Rather, we should use it to help us focus upon this idea — that Avinu Sh’bashamayim, our Father in Heaven, has asked for our presence together with him, one last day.
All the other Yamim Tovim are connected somehow to this physical world. Shemini Atzeres is not, according to the Nesivos Shalom. It is for this reason that the reason for its observance is not stated directly in the Torah.
Shemini Atzeres alludes to the time of future salvation where all the tikkunim, the fixing and the tweaking of the world’s Divine Service, will be perfected.
The Talmud tells us (Sukkah 47a) that in Chutz La’Aretz, outside of Israel, on the eighth day which is Shmini Atzeres we sit in the Sukkah, but we do not recite a blessing upon the sitting. Although this is the ruling in the Talmud as well as in the Rambam, the Tur, and the Shulchan Aruch, some Chasidim have the custom to eat in the home on Shmini Atzeres and not in the Sukkah. This has been a matter of great debate among families with mixed Chasidic and Ashkenazic legacies.
There is also a custom mentioned by the Ramah (OC Siman 667) to bid goodbye to the Sukkah at the last time one sits in it and to recite a special Yehi Ratzon. The Mishna Brurah (667:3) mentions in the name of the Shla that those who strive toward constant improvement have the custom to kiss the Sukkah upon entry and exit.
While in Eretz Yisrael Simchas Torah and Shmini Atzeres are celebrated on the same day, in chutz laAretz Simchas Torah is on the second day of Yom Tov. The completion of the cycle of Torah reading is celebrated.
After Maariv, pesukim are handed out by the gabbai to be recited aloud. The first passuk is “Atta hareisa.” Then the sifrei Torah are removed from the aron and everyone joins in seven hakafos (circling), with singing and dancing. In some communities the Torah is read at night as well.
In the morning, there are another seven hakafos before the Torah is read. It is the custom for every man to receive an aliyah to the Torah on Simchas Torah. Indeed, even the children receive an aliyah which is called “Kol HaNe’arim.” A large tallis is held over them as the Torah is read.
There are two important aliyos that are given on this day: “Chassan Torah,” to who completes the Torah, and “Chassan Bereishis” to who begins it anew. The “Bereishis Torah” is read immediately to show that we are never, Heaven forbid, “done” with Torah. A special prayer is recited by the gabba’im for the people receiving these aliyos before they are called up.