Rejoice greatly, O daughter of Sion, shout for joy, O daughter of Jerusalem: BEHOLD THY KING will come to thee, the just and saviour: he is poor, and riding upon an ass, and upon a colt the foal of an ass. Zechariah 9:9
Palm Sunday celebrates Christ's triumphal entry into Jerusalem.
As he came to the city of Jerusalem, he began to weep over Jerusalem as told in Luke 19 -
41 And when he was come near, he beheld the city, and wept over it,
42 Saying, If thou hadst known, even thou, at least in this thy day, the things which belong unto thy peace! but now they are hid from thine eyes.
Riding upon a donkey showed Christ not as the king of war, but rather the Prince of Peace. As Jesus entered the city, the people celebrating their King's arrival threw down their cloaks and palm branches, singing Psalm 118 -
Blessed is he who comes in the name of the Lord
It was custom of the time to cover the path of someone coming who was worthy of the highest of honors - hence the people were giving Christ Jesus this honor. King Solomon had palm branches carved into the walls and the front doors of the Temple (1 Kings 6:29). Palm fronds specifically are one of the four species carried for Sukkot, as prescribed for rejoicing in Leviticus 23:40:
40 On the first day you are to take branches from luxuriant trees—from palms, willows and other leafy trees—and rejoice before the Lord your God for seven days.
The palm also had significant meaning to the other cultures present in Jerusalem - the Egyptians believed it to signify Eternal Life and the Greco-Roman culture used the palm branch as a symbol of triumph and victory.
In the Roman Catholic Church, palm fronds are blessed with Holy Water outside the church building during the Blessing of the Palms. After being blessed, the palm fronds are then processed solemnity into the church. The vestments worn are to be Red indicating the supreme sacrifice Christ was making as he entered the city.
In Belgium, specifically Hoegaarden, one of the last remaining Palm Sunday processions still takes place each year, with Twelve men acting as Apostles carrying a wooden statue of Christ. Bulgarians celebrate Palm Sunday as Tsvetnitsa, comonly called Flower's Day. People with names of flowers (Lily) celebrate the day as their name day in addition to the Christian aspects of the day.
In India, flowers are strewn across the sanctuary during the reading of the Gospel followed by the people repeating three times:
"Hosanna! Blessed is he who is come and is to come in the name of the Lord God"
Afterwards, as the congregation shouts Hosanna one final time, the flowers are thrown across the entire building.
Polish towns and villages organize artificial palm building competitions, with some of these palms reaching over 90 feet tall. A Spanish custom is to wear something new (clothing or shoes) with an old saying being:
Domingo de Ramos, quien no estrena algo, se le caen las manos
The Welsh and English celebrate the day as Sul y Blodau or Flowering Sunday, a day to clear and clean graves before placing flowers.