You may have heard Sukkot referred to as a 'composite of holidays'. What does this mean?
Sukkot itself is celebrated for seven days (in which case, Shemini Atzeret and Simchat Torah are combined) in Israel and liberal Jewish communities outside and for eight days for other Jewish communities. The seventh day is known as 'Hoshanah Rabbah' -the great 'save us now' (sadly, that's the best translation I could come up with). A hoshanah is a liturgical poem asking Gd to save us. Some have described the day as a minor Yom Kippur where we are judged by our merit (and hence the reason the Yom Kippur greeting 'g'mar chatimah tovah' is used through this day). It is mentioned in Psalms 118:25. One of the traditions is to beat our Sukkot willows into the ground (symbolic of ridding ourselves of our transgressions) while making seven hakafot/processions with lulavim and etrogim around the sanctuary. Prayers for rain, critical for the agricultural society of our ancestors and for all of us today, are a highlight and remain included in our Amidah until Pesach. Hoshanah Rabbah is indeed practiced today.
Here is a description of the early practice of Hoshanah Rabbah from the Mishnah (Sukkah 4.5):
"There was a place below Jerusalem called Moza. They went down
there and gathered tall branches of aravot (willows) and then they
came and stood them up at the sides of the altar, and their tops were
bent over the altar. They then sounded a teki'ah [long blast],a teru'ah
[staccato blast] and again a teki'ah. Every day they went round the
altar once, saying, 'O Lord, save us, O Lord, make us prosper"
Shemini Atzeret is the day following (this year, Friday night and Shabbat. It is a yom tov in its own right). It means 'eighth day of assembling' and is no longer Sukkot, but an 'add-on' day. Rashi (the great medieval scholar) teaches that it is as if G-d says to us: we've been together through this long holy day season and I don't want you to go just yet - stay a little longer. It is difficult for Me to part from you. In Israel and liberal Jewish communities, it is combined with Simchat Torah, yet not originally intended to be so as Shemini Atzeret is mentioned in the Tanakh, but Simchat Torah is not (it was likely adopted as a holiday celebrating the Torah reading cycle in the middle ages).
Simchat Torah means 'rejoicing with the Torah' (not to be confused with Shavuot when we received the Torah). It marks the transition in our Torah cycle reading/study when we read from the very last parasha in the Torah (Ve-zot ha-beracha, only read on Simchat Torah) and immediately start Bereshit over again to show that we never stop learning Torah. Hakafot/processionals with the Torah scrolls are celebrated and it is said that, as we dance with the Torah scrolls, we become the Torah's 'feet'. There is a special group Aliyah for children under Bar/Bat Mitzvah age.
Ve-zot ha-beracha opens with the blessing Moses gives to the Israelites just before his death and concludes with the end of Moses' life: "Never did there arise a prophet in Israel like Moses - whom G-d singled out face to face for the various signs and portents that G-d sent him to display in the land of Egypt against pharaoh and all his courtiers and his whole country, and for all the great might and awesome power that Moses displayed before all Israel."