Palm Sunday commemorates the entrance of Jesus into Jerusalem (Matthew 21:1-9), when palm branches were placed in his path, before his arrest on Holy Thursday and his crucifixion on Good Friday.
This is the day Jesus triumphantly entered Jerusalem riding on a donkey — the beginning of Holy Week — the last week of Lent and the week observed by many Christian churches as a time to commemorate and enact the suffering and death of Jesus.
Palm Sunday is known as such because the faithful will often receive palm fronds which they use to participate in the reenactment of Christ's arrival in Jerusalem. In the Gospels, Jesus entered Jerusalem riding a young donkey, and to the lavish praise of the townspeople who threw clothes, or possibly palms or small branches, in front of him as a sign of homage. This was a customary practice for people of great respect.
Palm branches are widely recognized symbol of peace and victory, hence their preferred use on Palm Sunday.
The use of a donkey instead of a horse is highly symbolic, it represents the humble arrival of someone in peace, as opposed to arriving on a steed in war. A week later, Christ would rise from the dead on the first Easter.