Holy Week begins today. The entry into Jerusalem is seen as the prophetic fulfillment of Zacharias 9: 9-10 :
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. And I will destroy the chariot out of Ephraim, and the horse out of Jerusalem, and the bow for war shall be broken: and he shall speak peace to the Gentiles, and his power shall be from sea to sea, and from the rivers even to the end of the earth.
Before the Mass is the Blessing of the Palms, which includes an Antiphon, Psalms, and Gospel reading. Then comes the Procession with hymns, when we carry the palms either around the church or outside, weather permitting, and then the Mass, during which there is a very long reading sung in three parts by three deacons (or priest and deacons such as the case may be) - a long recitation of the Passion, including Matthew 26: 36-75 and Matthew 27: 1-60.
Carrying palms (or olive or willow branches, if palms are not available) in procession originates in the Old Testament where it was not only approved but commanded by God at the very foundation of the Old Testament religion. In the autumn of the year, after the harvest, when the people gathered for the Feast of Tabernacles God said in Leviticus 23: 40:
And you shall take to you on the first day the fruits of the fairest tree, and branches of palm trees, and boughs of thick trees, and willows of the brook: And you shall rejoice before the Lord your God.
Again we read of palms in the II Machabees 10: 6-8:
And they kept eight days with joy, after the manner of the feast of the tabernacles, remembering that not long before they had kept the feast of the tabernacles when they were in the mountains, and in dens like wild beasts. Therefore they now carried boughs and green branches and palms, for him that had given them good success in cleansing his place. And they ordained by a common statute, and decree, that all the nation of the Jews should keep those days every year.
And in the seventh chapter of Revelation, we find that those who were "sealed" are seen by John carrying palms:
Revelation 7: 9-10:
After this, I saw a great multitude, which no man could number, of all nations and tribes and peoples and tongues, standing before the throne and in sight of the Lamb, clothed with white robes, and palms in their hands. And they cried with a loud voice, saying: Salvation to our God, who sitteth upon the throne and to the Lamb.
The palms are blessed before the High Mass today.