This is what it says in the Bible:
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 God for seven days. (English Standard Version)
So this is what we do. We take the three branches of trees mentioned (palm fond/lulav, hadass/myrtle and aravah/willow and we bind them together. We hold the fruit of splendid trees (interpreted as an etrog/citron) together with the bound branches and we shake it all about. I'm not being funny. We say a blessing and shake towards the four compass points and up and down, to show that God is everywhere, whilst praying for His mercy and forgiveness. (The Book of Life that we want to be inscribed in for another year is not closed and sealed until the last day of Sukkot.) We shake the lulav every day of the festival of Sukkot (but, as the song says, not on Shabbat).