The origin of Palm Sunday is documented in The Gospel of Matthew 21:8-9. On the Sunday before His crucifixion, the scripture reads "And a very great multitude spread their garments in the way; others cut down branches from the trees, and strawed them in the way. And the multitudes that went before, and that followed, cried, saying, Hosanna to the Son of David: Blessed is he that cometh in the name of the Lord; Hosanna in the highest."
Any left over palms from the Palm processions are burned and the ashes used in the subsequent year's Ash Wednesday. In countries without palm trees, branches of different trees are often used. Palm Sunday marks the beginning of Holy Week too. There are events occurring throughout the week culminating with Good Friday and Easter Sunday.