From the Shemini Atzeret Wikipedia page:
Shemini Atzeret is a Jewish holiday. It is celebrated on the 22nd day of the Hebrew month of Tishrei in the Land of Israel, and on the 22nd and 23rd outside the Land, usually coinciding with late September and/or early October. It directly follows the Jewish festival of Sukkot which is celebrated for seven days, and thus Shemini Atzeret 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, this is further complicated by the additional day added to all Biblical holidays except Yom Kippur. The first day of Shemini Atzeret therefore coincides with the eighth day of Sukkot outside of the Land of Israel, leading to sometimes involved analysis as to which practices of each holiday are to apply.
The celebration of Simchat Torah is the most distinctive feature of the holiday, but it is a later rabbinical innovation. In the Land of Israel, the celebrations of Shemini Atzeret and Simchat Torah are combined on a single day, and the names are used interchangeably. In the Diaspora, the celebration of Simchat Torah is deferred to the second day of the holiday. Commonly, only the first day is referred to as Shemini Atzeret, while the second is called Simchat Torah.
Karaite Jews and Samaritans also observe Shemini Atzeret, as they do all Biblical holidays. However, it may occur on a different day from the conventional Jewish celebration, due to differences in calendar calculations. Karaites and Samaritans do not include the rabbinical innovation of Simchat Torah in their observance of the day; and do not observe a second day (of any holiday) in the Diaspora.