Why are people waving palm branches?
The Jewish eight day celebration of the harvest, known as Sukkoth, and also called the Festival of the Tabernacles or Booths, was probably the most popular holiday among the Jews in the first century. In the observance of Sukkoth, worshippers processed through Jerusalem and in the Temple, waving a bunch of leafy branches made of willow, myrtle and palm. As they waved these branches in that procession, the worshippers recited words from Psalm 118, the psalm normally used at Sukkoth. Among these words were "Save us, we beseech you, O Lord." Save us in Hebrew is hosianna or 'hosanna'. This is typically followed by "Blessed is the one who comes in the name of the Lord. (Psalm 118:25-6)."
Although the Passover is too early for leafy branches (except palms), Mark 11:8-9 says (NAB), "Many people spread their cloaks on the road, and others spread leafy branches that they had cut from the fields. Those preceding him as well as those following kept crying out: 'Hosanna! Blessed is he who comes in the name of the Lord!'." The Gospels of Matthew and Luke more or less follow Mark, but John's Gospel corrects this to say 'palm branches', creating our modern tradition of Palm Sunday.