In Israel, Shemini Atzeret is the day after the seventh day of Sukkot. This day is also the holiday Simchat Torah. Outside of Israel, where extra days of holidays are held, only the second day of Shemini Atzeret is considered Simchat Torah. Shemini Atzeret is Tishrei 22 and 23, while Simchat Torah is Tisheri 23.
These two holidays are commonly thought as part of Sukkot, which is incorrect. Shemini Atzeret is a holiday in its own right and does not involve some of the observances of Sukkot.
Shemini Atzeret literally means “the assembly of the eighth (day).” According to rabbinic literature, our Creator is a host who invites us as visitors for a limited time. When the time comes for us to leave, He has enjoyed Himself so much that He asks us to stay another day.
Simchat Torah literally means “rejoicing in the Torah.” This holiday marks the completion of the annual cycle of weekly Torah readings. Each week in the synagogue, Jewish people publicly read a few chapters of the Torah, starting with Genesis chapter 1 through Deuteronomy 34. On Simchat Torah, the last Torah portion is read, then proceed immediately to Genesis chapter 1, a reminder that the Torah is a circle and it never ends.
The completion of the readings is a time of great rejoicing. There are processions around the synagogue carrying Torah scrolls with plenty of high-spirited singing and dancing. In some synagogues, confirmation ceremonies or ceremonies marking the beginning of a child’s Jewish education are held at this time.