In the Gospels, Jesus' entry into Jerusalem takes place a week before His Resurrection.
"The Coming of Zion's King – See, your king comes to you, righteous and victorious, lowly and riding on a donkey".
The prophets of the Old Testament had foretold the coming of Jesus, and by riding in on a donkey, Jesus was fulfilling the prophesy. He was saying that he was the King of Israel. That is why it was seen as a threat by the priests.
Many people celebrated His arrival, and laid down their cloaks in front of Him. They also laid down small branches of trees.
In the Eastern tradition a donkey is an animal of peace, and was cheap to buy, whereas the horse was expensive and the animal of war. Jesus' entry to Jerusalem said He was entering as the Prince of Peace, not as a warring king.
In many lands in the ancient Middle East, it was customary to cover the path of a dignitary, much as we might roll out the red carpet today.
The palm branch was a symbol of victory in the Roman Empire. In ancient Egypt, the palm was carried in funeral processions and represented eternal life. The palm branch later became a symbol of Christian martyrs and their victory over death.