What are the similarities between Passover and the Eucharist?

Great question! Actually, the Christian (primarily Catholic) tradition of celebrating the Eucharist developed out of Passover traditions. Jesus of Nazareth was raised in a Jewish society and even grew up to be a Rabbi. As such, he celebrated the holiday of Passover. This holiday is celebrated in remembrance of when God freed the Jewish people from enslavement in Egypt. Moses had demanded freedom for the Jewish people from Pharaoh many times, but he would not let them go. God sent swarms of frogs and sickness on the Egyptian people as a warning, but still Pharaoh would not free the Jewish slaves. The final plague God sent upon the Egyptians was that all of the first-born children would die. However, the Jewish people and their children were protected--God "passed over" their homes. It was after this plague that Moses led the Jewish people out of Egypt and to their freedom.

Jesus of Nazareth would have celebrated this holiday in springtime in remembrance of his heritage as well as part of his duties as a Rabbi. While Passover traditions vary from place to place, and even from family to family, all Passover celebrations involve a meal. Before his crucifixion, Jesus gathered with his disciples to share a Passover meal. Jesus knew that in the coming days, he would be crucified and die. It was at this meal that he told his disciples to eat some bread and drink some wine in remembrance of him. He said that his followers should remember his life as well as his message to love one another (even your enemies), and to accompany this remembrance by eating some bread and wine. One of the early doctrines of the Christian faith was that people continued to celebrate the life of Jesus by getting together to share some bread and wine.

Today, the Eucharist (from the Greek for "giving thanks") is celebrated regularly by Catholics and some other Christian denominations. The Eastern Orthodox Church continues to use bread for their celebration of the Eucharist, but most congregations who celebrate this Sacrament use a little wafer made of wheat flour. The Eucharist is considered a symbol for and manifestation of the sacrifice Jesus of Nazareth made when he was crucified. As opposed to Jesus' celebrations of the Passover meal, the Eucharist may be celebrated daily (though most do it weekly) in remembrance of Jesus and his message. 

Both the Passover meal and the modern Eucharist are traditions focusing on God's covenants with man--the promises God and man made to one another about their responsibilities in their relationship. The Passover celebration has to do with the founding of Judaism and God's "Ten Commandments" given through Moses. In the Christian tradition, when people failed to live up to God's original commandments, Jesus was sent to establish a new covenant. He led by example and asked others to fulfill their duty to God by simply loving one another. Celebrating Passover and the Eucharist are both ways of reminding oneself and others of the duties and relationship between God and man.