We also wave the four species. This based on a biblical commandment:
Leviticus 23:40 And you shall take on the first day the fruit of splendid trees, branches of palm trees and boughs of leafy trees and willows of the brook, and you shall rejoice before the LORD your G-d for seven days. English Standard Version
We use the lulav (date palm frond), etrog (citron), Hadass (myrtle branch with leaves), and aravah (willow branch with leaves). We make a blessing each day when we take up the four species, and an additional blessing on the first day to celebrate the first time we take them during the holiday. We then shake the species three times in each of six directions...forward, right, back, left, up, down. We don't bless the objects (the four species for example), but we bless G-d either for commanding or making it possible for us to do something or for supplying us with something we need (such as food, life, health) and creating a wonder that we might enjoy (such as a rainbow, or a new fruit (new to the season) or for enabling us to celebrate a holiday or other occasion.
We also wave them during the Hallel (psalms recited during holidays) during the morning service. Finally we have a parade holding the spices in the synagogue following the prayer leader each of the seven days of the holiday. The picture was taken on Hoshana Raba 2013 at Temple Israel in Sharon MA. Yours truly happened to look up in time to smile at the photographer, Benjamin Maron.
The last day is Hoshana Raba, and on that day we parade seven times. All the while we ask G-d to save us. Cantor Steve Dress pointed out that we are also praying for rain. Rain is essential since we cannot live without water. It also refers to Torah, and and that is something a religious Jew cannot live without. Hoshana Raba ends the season that began with Rosh Hashana- the religious new year. This is the season when we ask G-d and our fellow man to forgive and sins we may have committed...hopefully lightening our load so we can start the new year with a clean slate.
Heaven decides if we will live another year, and Hoshana Raba is the last day for us to attempt to rid ourselves of the previous years deeds. At the end of the seventh procession, we take our arava (willow branches) and beat them on the ground until all the leaves fall off, symbolically getting rid of our sins. After this we start celebrating as Shimini Atzeret leads to Simchat Torah where we celebrate the giving of the Law on Sinai.
The video on the left shows the blessing over the species, and then an Etrog box gallery. Etrogs are often sold in boxes, and people can purchase or make decorated etrog boxes. Mine has a velcro closure and some cushioning foam on the inside. I wouldn't go anywhere without it...or at least my etrog wouldn't.