Palm Sunday is the sixth Sunday of Lent and marks the start of Holy Week. It commemorates the triumphal entry of Jesus Christ into Jerusalem on a donkey the week before the Resurrection. It is an event which features in the Gospels of Matthew, Mark, Luke and John, and here it is from John:
On the next day much people that were come to the feast, when they heard that Jesus was coming to Jerusalem, took branches of palm trees, and went forth to meet him, and cried, 'Hosanna: Blessed is the King of Israel that cometh in the name of the Lord.' And Jesus, when he had found a young ass, sat thereon; as it is written, 'Fear not, daughter of Sion: behold, thy King cometh, sitting on an ass's colt'. (John 12: 12-15, King James Version)
In Tudor times the priest would read out the story and then bless branches of greenery to be used in processions. Apparently, a wooden donkey on wheels was used in some processions!