Shemini Atzeret means “Eighth Day of Assemby” and it’s a holiday that confuses many people. In some ways, it’s the eighth day of Sukkot — the capstone of the week-long harvest festival. In others, it is an independent holiday.
In 2019, Shemini Atzeret begins at sundown on Monday, October, 20th and ends at sundown on Tuesday, October 21st.
The medieval exegete Rashi understood Shemini Atzeret through a parable. Imagine a king who had invited his children to dine with him for seven days, but after a week begged them to stay one day longer. In this parable, God is the king and the Jews are the children. The banquet is Sukkot (celebrating the bounty of the harvest) which is stretched on for one extra day, Shemini Atzeret. It’s a rather lovely description of God’s relationship to the Jewish people.
For a more detailed explanation about whether Shemini Atzeret is part of Sukkot or not, and the practices that overlap with Sukkot and those that do not, see this article.
Today in Israel, Sukkot is only seven days long, and Shemini Atzeret follows directly after. In the Diaspora, where Sukkot is observed for eight days, Shemini Atzeret coincides with the final day of Sukkot. (Note: For Diaspora Jews, particularly Reform Jews, who keep only seven days of Sukkot, Shemini Atzeret follows Sukkot.)
In sum, all Jews will observe Shemini Atzeret on Tuesday, October 21st this year, the difference is that for some it will also be the eighth day of Sukkot.
The day before Shemini Atzeret, the seventh day of Sukkot, is called Hoshanah Rabbah — falling this year on Monday, October 20th. The day following Shemini Atzeret is Simchat Torah — this year on Wednesday, October 22nd. However, some communities combine Shemini Atzeret and Simchat Torah and will observe both on Tuesday, October 21st.