As we come to the end of the High Holidays, we have two days left. Of course, as you might expect, in Israel, both of these days are celebrated on a single day.
Shemini Atzeret means the Assembly of the Eighth [Day]. It occurs the day after the seventh day of Sukkot. While its name might imply that it is part of Sukkot, it is a separate holiday. However, because of the tradition of adding a day to many observances outside of Israel, it can be thought of as coinciding with the last day of Sukkot. Certain practices of Sukkot are not carried over for Shemini Atzeret, like sleeping in the Sukkah and saying the prayer with the lulav and etrog.
Rabbinic tradition tells us that Shemini Atzeret is when the world is judged for rainfall, which is of special concern in Israel, and we recite prayers for rain on the day. It is also understood as a day highlighting the relationship between the Jewish people and G-d.
Simchat Torah is a celebration of the Torah. Each year, the Torah (the five books of Moses) is read in the synagogue in portions. On Simchat Torah, the last portion is read immediately followed by the very beginning of Genesis. As many people as possible are called for an aliyah, to recite a blessing for the Torah reading. It is a joyous celebration accompanied by dancing and singing when the ark is opened and the Torahs are revealed and marched around the synagogue.
In reformed synagogues, the Torah portions are often divided up so the entire Torah is read every three years. Nevertheless, the holiday is also celebrated each year.
Both holidays begin at sundown and last until nightfall of the next day, Shemini Atzeret on October 7th and Simchat Torah on October 8th.