The holiday of Sukkot is followed by an independent holiday called Shemini Atzeret. In Israel, this is a one-day holiday; in the Diaspora it is a two-day holiday, and the second day is known as Simchat Torah. This holiday is characterized by utterly unbridled joy, which surpasses even the joy of Sukkot. The joy reaches its climax on Simchat Torah, when we celebrate the conclusion—and restart—of the annual Torah-reading cycle.
These two days constitute a major holiday, when most forms of work are prohibited. The special joy of this holiday celebrates the conclusion—and restart—of the annual Torah-reading cycle On the preceding nights, women and girls light candles, reciting the appropriate blessings, and we enjoy nightly and daily festive meals, accompanied by kiddush. We don’t go to work, drive, write, or switch on or off electric devices. We are permitted to cook and to carry outdoors (unless it is also Shabbat).
The first day, Shemini Atzeret, features the prayer for rain, officially commemorating the start of the Mediterranean (i.e., Israeli) rainy season, and the Yizkor prayer (supplicating G‑d to remember the souls of the departed).
We no longer take the Four Kinds, and we no longer mention Sukkot in the day’s prayers; in the Diaspora, however, we do still eat in the sukkah (but without reciting the blessing on it).
The highlight of the second day, Simchat Torah (“The Joy of the Torah”), is the hakafot, held on both the eve and the morning of Simchat Torah, in which we march and dance with the Torah scrolls around the reading table in the synagogue. (In many synagogues, hakafot are conducted also on the eve of Shemini Atzeret.)
On this joyous day when we conclude the Torah, it is customary for every man to take part in the celebration by receiving an aliyah. The children, too, receive an aliyah!
After the final aliyah of the Torah, we immediately begin a new cycle from the beginning of Genesis (from a second Torah scroll); this is because as soon as we conclude studying the Torah, G‑d’s infinite wisdom, on one level, we immediately start again, this time to discover new and loftier interpretations.
(In the Land of Israel, the celebration and customs of these two days are compressed into one day.)