In general, the palm is a symbol of victory and triumph. It is associated with the rejoicing that comes with victory. Thus saints, especially martyrs, are often depicted carrying the palm of victory—they have triumphed over sin and won the victory of heaven.
All the Gospels recall the triumphant entry of Jesus into Jerusalem before his passion and death. The Gospels tell us that the crowds lined the road welcoming Jesus to the city. And they laid branches from the trees or reeds on the road before Jesus. John recalls, "...they took palm branches and went out to meet him, and cried out, ‘Hosanna! Blessed is he who comes in the name of the Lord...’"(12:13).
In the first part of the liturgy on Passion (Palm) Sunday we commemorate and reenact Jesus’ triumphant entry into Jerusalem. Palms are blessed and given to those present to carry in procession.
In the blessing of palms the priest prays, "Today we honor Christ our triumphant King by carrying these branches. May we honor you every day by living always in him."