Shemini Atzeret

Shemini Atzeret ( – "the Eighth [day] of Assembly") is a Jewish holiday of the (Northern Hemisphere) autumn. In Israel, the date of the holiday is 22 Tishrei on the Hebrew calendar. Outside of Israel, it is two days long, on 22-23 Tishrei. On the Gregorian calendar, Shemini Atzeret comes in late September or early or middle October.
The holiday is best known for the celebration of Simchat Torah (שִׂמְחֵת תּוֹרָה), "Rejoicing of the Torah", when Jews finish the annual cycle of reading the whole Torah out loud in synagogue. Outside Israel, "Simchat" Torah is celebrated on the second day of the holiday. Most people outside Israel use the name "Shemini Atzeret" for the first day only. They call the second day "Simchat Torah" because of this celebration. In Israel, the one day of the holiday is called by both names.
In some ways, this holiday is part of the holiday of Sukkot. The "shemini" ("8th") is the "8th day" of Sukkot. In other ways, this holiday is a separate holiday on its own, and not part of Sukkot. 
Shemini Atzeret is an important Jewish holiday. Like on Shabbat and other important Jewish holidays, work is not allowed on Shemini Atzeret.
Sources in the Torah.
The holiday of Shemini Atzeret is in the Torah (Pentateuch) twice, at and at . The Torah does not say much about the holiday, though.
Shemini Atzeret and Sukkot.
The Torah says that this holiday falls on the eighth day of Sukkot. However, in the same paragraph, the celebrations of Sukkot are described as lasting "seven" days, not eight. The Talmud writes about this apparent(def. 2) problem. At the end of the section, the Talmud says that at the same time,
The laws and customs of Shemini Atzeret show both pieces of the holiday.
How Shemini Atzeret is celebrated.
Some Jewish holidays have special rituals, like the Seder of Passover or the sukkah, lulav and etrog of Sukkot. In the Torah, there are no rituals like that for Shemini Atzeret. But starting at the time of the Talmud, Shemini Atzeret became the holiday for finishing the annual reading of the Torah in the synagogue. The celebration of that—called Simchat Torah— has become the main feature(def. n2) of the holiday.
Simchat Torah.
In synagogues, one section of the Torah is read on every Shabbat of the year. The last section, from Deuteronomy, is read on the holiday of Shemini Atzeret. Over the last 1000 years, the reading of the last section has become a big celebration called Simchat Torah, or "Rejoicing of the Torah". This celebration is so important that the whole day of the celebration is called "Simchat Torah." In Israel, this celebration happens on the single day of the holiday, 22 Tishrei. Outside of Israel, it happens on the second of the two days, 23 Tishrei.
History.
The name "Simchat Torah" does not appear in the Torah or the Talmud. The Talmud says that the final section of the Torah is read on Shemini Atzeret. But it does not use the name Simchat Torah, and it does not describe a special celebration. The first reference to any kind of celebration comes from the perioddef. "n1" of the "Geonim" (7th-11th centuries CE). That source mentioned the custom of dancing with the Torah, which is still a custom today.
In modern times, Simchat Torah has become a day to show Jewish pride in public. 
Simchat Torah evening.
Simchat Torah evening is a wikt:festive and child-friendly holiday. After the regular holiday evening prayers, all the Torahs are taken from the wikt:ark. Members of the synagogue march or dance around the synagogue seven times with the Torahs. Each time around the synagogue, the people pray "Save us! Answer us on the day we call!" Sometimes the dancing moves from the synagogue to the street outside. Sometimes the dancing can go late into the evening, with extra songs and prayers added. One very common custom is for people—especially children—to march or dance with flags, sometimes with candles or apples on top. At the end of the dancing in most synagogues, a piece of the last Torah portion in Deuteronomy is read in public. In Orthodox and Conservative synagogues, the very end of Deuteronomy is not read at night. But in Reform synagogues, it sometimes is.
Simchat Torah morning.
After the regular holiday morning prayers, the dancing and marching of the previous night are repeated. After that, every wikt:eligible adult member of the synagogue is called to the Torah for an "aliyah" (honor), says blessings, and has several verses of the Torah read. Depending on the size of the synagogue, this can take a long time. Different synagogues have different ways to do this:
Once all eligible adults have received an honor, the celebration comes to its highest point. The last three honors go to distinguished members of the synagogue:
For all of these honors, the person honored stands under a "tallit" (prayer ) held open like the "ḥuppa," or , in a Jewish wedding.
"Hakafot Shniyot".
In Israel, it has become common to add an extra night of celebration to Simchat Torah. This is called "Hakafot Shniyot" (Second Circuits). It happens on 23 Tishrei, the same night as Jews in the rest of the world celebrate Simchat Torah. This is a way for Jews around the world to show Jewish pride together. Because the holiday of Shemini Atzeret is over after one day in Israel, holiday restrictions do not apply. So "Hakafot Shniyot" can include things not normally allowed on Shabbat and Jewish holidays (like bands and photography).
Practices from Sukkot.
The Torah states that it is a requirement for Jews to be happy on Sukkot. The Talmud writes that this requirement lasts for eight days—the seven days of Sukkot "and" Shemini Atzeret. As part of this, Hallel is said in synagogue during the morning prayers of Shemini Atzeret.
In Israel, this is the only practice from Sukkot that continues into Shemini Atzeret. In every other way Shemini Atzeret is a separate holiday with separate practices.
Outside Israel, this is complicated by the Jewish law of adding an extra day to every Biblical Jewish holiday except Yom Kippur. The holiday of Sukkot, which lasts seven days in Israel lasts eight days outside Israel. But the eighth day is also the first day of the separate holiday of Shemini Atzeret. Sometimes the rules for the two holidays conflict. Figuring out the conflicts can be complicated. But the general rule is that 
In practice, what happens "only outside of Israel" is that
None of this happens on the second day of Shemini Atzeret (Simchat Torah), because it is no longer an extra day of Sukkot.