Shemini Atzeret is a Jewish holiday also known as The Eighth (Day) of Assembly, which takes place the day after the Sukkot festival, where gratitude for the fall harvest is deeply internalized. Simchat Torah marks the end of the weekly readings of the Torah. The Torah is read from chapter one of Genesis to Deuteronomy 34 and then back to chapter one again, in acknowledgement of the words of the Torah being a never-ending cycle.
Why is it important?
Shemini Atzeret directly follows the Jewish festival of Sukkot which is celebrated for seven days, and thus is literally the eighth day. It is a separate—yet connected—holy day devoted to the spiritual aspects of the festival of Sukkot. Part of its duality as a holy day is that it is simultaneously considered to be both connected to Sukkot and also a separate festival in its own right.
Outside the Land of Israel, an additional day added to all Biblical holidays, therefore the first day of Shemini Atzeret coincides with the eighth day of Sukkot, leading to sometimes involved analysis as to which practices of each holiday are to apply.