There are three distinct and special celebrations at the end of Sukkot. The first of them, Hoshana Rabbah, is today.
Hoshana Rabbah -- "The Great 'Save-Us!'" -- is the seventh day of Sukkot and a minor holiday in its own right. On this day, traditionally, we make seven circuits of our sanctuaries with our lulavim and our Torah scrolls while reciting prayers called Hoshanot which ask God to bring healing and salvation. Seven is a number with spiritual significance in Judaism: seven days of the week, the seven "lower" (accessible) sefirot (aspects of God), the seven ancestral figures (some invite seven men and seven women) welcomed into the Sukkah as ushpizin (holy guests) -- and now on the seventh day of this festival we circumnambulate our sanctuaries seven times, singing and praying. There's also a very old custom of taking the willow branches from our lulavim and beating them against the ground; the falling willow leaves are an embodied prayer for rain. (For more on that: The Ritual of Beating the Willow.)
Even if you're not dancing or processing around a sanctuary with branches and Torah scrolls, reading some hoshanot and reflecting on their meaning is a lovely observance of Hoshana Rabbah. I like the ones written by my teacher Rabbi Zalman Schachter-Shalomi, which online here at the Reb Zalman Legacy Project blog, and also here at the Shalom Center (with commentary from Rabbi Arthur Waskow below the hoshanot themselves.) And here's a brief excerpt from a translated hoshana, provided by Rabbi David Seidenberg of NeoHasid.org.
Shemini Atzeret -- "The Pausing of the Eighth Day" -- is the eighth day of Sukkot. Yes, Sukkot is a seven-day holiday, but tradition tells us that on the eighth day, God (Who has been so enjoying spending time with us in our sukkot) says "awww, do you really have to go? Can't you linger a little longer?" So we stay in our sukkot for one more day, one more chance to engage in intimate connection with Shekhinah, the immanent and indwelling Presence of God. This is a day for spaciousness, a day of pausing, a day to celebrate the white space which cradles and contains all of the texts and teachings and observances of the holiday season now ending. The Days of Awe and Sukkot are a dense and busy time, full of obligations and sermons and teachings; Shemini Atzeret is a chance to pause, to take a breath, to receive the blessings of stillness.
This day is a hinge-point in our liturgical year between the summer season and the winter season. On Shemini Atzeret, we recite special prayers for rain, and we enter into the liturgical winter-season when our Amidah contains a one-line prayer for rain every day instead of the summertime one-line prayer for dew. I've written a contemporary prayer for rain which can be read / davened on this day, which you can find in the VR archives here; you might also enjoy my Sestina for Shemini Atzeret, which I wrote last year and which I still really like. Shemini Atzeret is the 22nd of Tishrei, which begins tonight at sundown and lasts through tomorrow (Monday).
So when is Simchat Torah? Well, it depends on who you ask. In Israel, Sukkot lasts for seven days; the seventh day is Hoshana Rabbah; and the 8th day is both Shemini Atzeret and Simchat Torah. Reform communities outside of the land of Israel also operate on this calendar. (I'm not sure about Reconstructionist communities; Jewish Renewal communities' practices vary.) In Diaspora, Orthodox and Conservative Jews move from 7 days of Sukkot (the final one being Hoshana Rabbah) to two days of Shemini Atzeret, the second of which is Simchat Torah, making the whole shebang a 9-day observance insted of an 8-day one. (In my local community we'll celebrate Simchat Torah on Monday night, in conjunction with the local college Jewish student group.)
It's possible to experience a kind of holiday fatigue at this moment in the year. Tisha b'Av, then the month of Elul, then Rosh Hashanah, the Ten Days of Teshuvah, Yom Kippur, a week of Sukkot...! But there's beauty and meaning in each of these three final days of this holiday season. I offer this blessing: may each of us find a point of access into the beauty and wisdom of Hoshana Rabbah, Shemini Atzeret, and Simchat Torah. May we connect with the gift of being able to ask for help in healing our world, the gift of holy pausing and sacred rest, and the gift of Torah, the story which never ends.