Shemini Atzeret and Simchat Torah
The holiday of Sukkot is called "the time of our joy!"
The final two days of Sukkot are, at least formally, a totally separate holiday.
The Torah itself makes this clear and unclear at the same time, defining Shemini Atzeret as the eighth day of Sukkot at the same time that it grants it independent existence in the festival calendar.
Simchat Torah literally means “the joy of Torah” and is the special name given to what would otherwise be the second day of Shemini Atzeret in the Diaspora, the day on which the annual cycle of Torah readings begins and ends.
Shemini Atzeret begins on Sunday evening, October 16th, and continues for a second day as Simchat Torah (for those in the diaspora) concluding on Tuesday night, October 18th.