The eighth day of the holiday of Sukkot is actually a separate holiday called Shmini Atzeret. We are not required to live in the sukkah or to take the lulav and etrog on Shmini Atzeret. However, outside the land of Israel the general custom is to eat in the Sukkah on the first day of Shmini Atzeret without the blessing. Shmini Atzeret is one day long in Israel and two days long outside of Israel.
On the second day of Shmini Atzeret we celebrate Simchat Torah. Simchat Torah, which means 'Celebration of the Torah', is a day of rejoicing on the completion of the cycle of Torah readings. Every Shabbat a section from the Torah is read. The entirety of the Torah, the Five Books of Moshe (Moses), is completed and restarted on Simchat Torah.
The congregation dances with the Torah scrolls and performs seven hakafos. Hakafos, or Revolutions, are performed as follows:
One member leads the other Torah-bearing members in a circuit around the synagogue while the congregation kisses the scrolls. The leader leads the congregation in reciting special prayers.
When the circuit is completed the congregation sings and dances until it is time to start the next hakafa.
This is done seven times.
Hakafos are performed on both the night and the day of Simchat Torah. Some communities celebrate hakafos on the first night of Shmini Atzeret as well.
On the day of Simchat Torah it is customary for every man to receive an aliyah (to be called up to the Torah).