Coming at the conclusion of Sukkot  are the 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 on two consecutive days.
Literally translated as the "Eighth Day of Assembly,” Shemini Atzeret is mentioned in the Bible, but its exact function is unclear. In Second Temple times, it appears to have been a day devoted to the ritual cleansing of the altar in the Temple. The Torah  describes the offering that was brought to the Temple on Shemini Atzeret, but once the Temple was destroyed  in 70 CE, nothing remained from the holiday’s ritual except liturgy requesting rain for a bountiful year.
From Shemini Atzeret until Passover, this short prayer for rain is inserted into the second blessing of the Amidah . These two holidays serve as the bookends of the agricultural season, at the beginning and end of the rainy season in Israel. It's also traditional to include the Yizkor , or memorial service, as part of the liturgy for Shemini Atzeret.
The lack of clear definition as to the purpose of Shemini Atzeret may have provided the impetus to celebrate it in conjunction with Simchat Torah , a celebration of the conclusion of one and the beginning of another annual cycle of readings from the Torah.
Simchat Torah probably originated during the medieval period and is characterized by joyful dancing with the Torah . The final portion of the Book of Deuteronomy  is read followed by the beginning of the Book of Genesis . In this manner, the annual cycle of Torah readings continues unbroken.
Simchat Torah conveys a clear message about the centrality of Torah in Jewish life. It is both a source of Jewish identity and a precious gift from God. Simchat Torah is the day on which the whole community gathers to come into direct contact with the Torah and to express joy in having received it.
Unlike many other holidays, the observance of both Shemini Atzeret and Simchat Torah are centered in the synagogue and community. On Shemini Atzeret, some still eat in the sukkah  (the traditional hut associated with the festival of Sukkot), but in contrast to Sukkot, no blessings are associated with that activity.
For more details about both holidays, visit myjewishlearning.com