The palm branches used in Palm Sunday services symbolize honor toward Jesus and the people's acknowledgment of him as the King. Palm Sunday is the day commemorating Jesus's triumphant entry into Jerusalem just days before his crucifixion.
All four Gospels relate the Palm Sunday event, but in the King James Version, only John mentions that the branches used to spread on the road before Jesus were palm branches. Matthew and Mark say "branches from the trees" and "branches off the trees" respectively, but without naming them. Luke omits the part about the branches altogether.
At the time of the Palm Sunday event, Jesus had recently raised Lazarus from the dead, and many Jews were believing in him. He was making his way to Jerusalem for the Feast of the Passover. The city was full of people, and many came out to meet him on his journey. These people, along with the disciples, spread their clothes and the palm branches as acknowledgment of his importance. They also shouted praises to him, calling him "the King" and the one "that cometh in the name of the Lord."
Palm branches had long had significance as symbols of victory. According to I Kings, Solomon had them carved into both the walls and the doors of the temple. In Revelation 7:9, the Bible says that people from every nation stood before Jesus wearing white robes and "holding palm branches in their hands."
Christian churches frequently use palm branches as part of the worship service on Palm Sunday to honor Jesus as Lord, much as the people did on that day approximately 2,000 years ago.