Shemini Atzeret is the final day of festivities at the end of Sukkot. It is followed by Simchat Torah, the holiday that marks the conclusion of the annual reading of the Torah and its beginning again–amid dancing, song, and celebration.
Coming at the conclusion of Sukkot are the two holidays of Shemini Atzeret and Simchat Torah. In Israel and among liberal Jews they are combined into one holiday on the day after the conclusion of Sukkot. Among more traditional Jews outside of Israel, they are observed separately from one another on two consecutive days. Shemini Atzeret means the “Eighth Day of Assembly,” while Simchat Torah means “Rejoicing in Torah.”
Material excerpted from http://www.myjewishlearning.com/