Palm Branches
Palms are sacramentals of the Church blessed on Palm Sunday (the Sunday before Easter) -- the day that commemorates Christ's entry into Jerusalem. Their purpose is to honor Christ's glory and Kingship, as did the inhabitants of Jerusalem who met Him, strewing palm branches on the street before Him.
 
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.

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 7th chapter of the Apocalypse, we see that those who were "sealed" are seen by John carrying palms:  
Apocalypse 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 Mass on Palm Sunday. Vested in red cope and standing ourside the church, the priest recites a short prayer, blessing the palms. Then ,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. A short homily may follow after the reading of the Gospel narrative. 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.
 
When Mass is finished we take them home and hang them over crucifixes or holy pictures.