While the Passover is one of God's appointed times, it is not listed in Scripture as one of the annual Sabbaths. It is a regular day of work—in fact, it is the preparation day for the first day of Unleavened Bread—but the first few hours, the evening portion of the day, is a significant memorial of two great events in God's plan for mankind: the death of the firstborn in Egypt and the sacrifice of our Savior, Jesus Christ.
The bulk of the instruction about the Passover is written in Exodus 12, and a great deal of it concerns the Old Testament ritual meal that was eaten on that evening. These details are types that were fulfilled in the sacrifice of Jesus Christ, so the New Testament church is no longer required to slay a lamb, since, as the apostle Paul writes, "For indeed Christ, our Passover, was sacrificed for us" (I Corinthians 5:7).
The New Testament Passover is modeled after the events that occurred during what is commonly known as the Last Supper, the Passover meal that Jesus ate with His disciples just before His arrest, trial, and crucifixion. Jesus began His instruction that evening with a command to wash one another's feet: "For I have given you an example, that you should do as I have done to you" (see John 13:1-17), and so we do.
The apostle Paul summarizes what happens next:
For I received from the Lord that which I also delivered to you: that the Lord Jesus on the same night in which He was betrayed took bread; and when He had given thanks, He broke it and said, "Take, eat; this is My body which is broken for you; do this in remembrance of Me." In the same manner He also took the cup after supper, saying, "This cup is the new covenant in My blood. This do, as often as you drink it, in remembrance of Me." (I Corinthians 11:23-25)
So, to commemorate His sacrifice—His broken body and His shed blood—by which He paid the penalty for human sin and consecrated the New Covenant (see Hebrews 9:11-28), Christians eat a little unleavened bread and drink a small amount of wine. In doing so, they acknowledge His sacrifice and rededicate themselves to their covenant with Him. It is clear from both the Old Testament and New Testament examples that only those who have made the covenant—Christ's disciples—are allowed to partake of the bread and wine, thus only baptized members should participate in this part of the service (see the principle in Exodus 12:43-49; also I Corinthians 11:27-29).
As Christ did after changing the Passover symbols, members of the church then listen to the words of Jesus' discourse to His disciples, which is found in John 13-17. Then, to close the service, they sing a hymn before concluding the solemn service (see Matthew 26:30; Mark 14:26).