Succot is a festival with two purposes - to celebrate the harvest, and to commemorate the wanderings in the desert of the people of Israel. The festival of Succot lasts for eight days in the Diaspora, and seven days in Israel. During these days, we recite the entirity of the Hallel.
As well as being a harvest festival, Succot also helps us remember the travelling of the Israelites through the desert - when they stopped at a place called Succot they made themselves temporary shelters, from where we get the Succah. We are commanded to buid ourselves these "booths" - temporary dwellings - in which we must live for the duration of the festival.
We wave the Arba Minim (four species) in shul, the species being a willow twig, myrtle twig, and palm branch, and Etrog (type of citrus fruit) bound together. There are many traditions as to why we use these species - one explanation is that they form an embodiment of the Jewish people - the lulav is the spine, the willow the eyes, the myrtle the nose, and the etrog the heart. We shake the minim in six directions - forwards, left, backwards, right, down, and up, to remind us that G-d is everywhere. Hospitality suring Succot is highly encouraged. One should always try to invite a poor person to the Succah for a meal, or invite someone who cannot build a succah.