Carrying palms (or olive or willow branches, etc., if palms aren't available) in procession goes way back into the Old Testament, where it was not only approved but commanded by God at the very foundation of the Old Testament religion. In the fall 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.
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.
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.
After a few verses from the New Testament, the priest reads the story of Christ's triumphal entry into Jerusalem the Sunday before His death, and about how the people put palms in the Savior's path and sang hosannas because, ironically, they expected a temporal victory by the One they thought would be the great military leader who would conquer the Romans..
Then we pray, begging God that we may in the end go meet Christ, that we may enter with Him into the eternal Jerusalem. The prayers ask God to bless the palms, that they may be sanctified and may be a means of grace and divine protection to those who carry them and treasure them with faith.
The palms are distributed to the people at the Communion rail. The priest will press the palm against your lips so you can kiss it, and then his hand. Alternatively, the palms may be handed out by the altar boys. In any case, Scripture and prayers follow, and then a procession of clergy, servers, and people through the church or outside around the church.
Some of these same palm branches are saved and burned the next year to make the ashes for the next Ash Wednesday -- the palms, which symbolize triumph, and the ashes, which sympbolize death and penitence, forming a great symbolic connection between suffering and victory.
The branches given to the faithful are held in the hand at the singing or reading of the Passion and the Gospel during the Mass, but when Mass is finished we take them home and hang them over crucifixes or holy pictures. Men will sometimes wear a piece of it in their hats or pin it to their lapels, and a piece should also be placed with one's sick call set.