In 1 Corinthians 11 the apostle Paul records for the Corinthians (and for us) Christ's institution of the Lord's Supper. Christ institutes for his followers a replacement for the Passover; a commemoration of the events that brought about salvation for God's people in the new covenant to replace a commemoration of the events that brought about salvation for God's people in the old covenant. The symbolizm is incredible. The Israelites sacrificed a lamb, and the lamb's blood painted on the door frame of the house was the sign of their salvation. Likewise, the Lamb was slain, and his blood is the sign of our salvation.
What is this new covenant? The old covenant, or the Mosaic covenant, was the covenant God established with the people of Israel in the wilderness after he delivered them from slavery in Egypt. God did not force them into the covenant, and they did not deserve this covenant. It was a covenant of grace. It was, however, a covenant that God's people could not keep. God was not surprised, however, and he planned a second covenant to replace the Mosaic covenant.
In Jeremiah 31:31-40, we find God's promise of a new covenant: "But this is the covenant that I will make with the house of Israel after those days, declares the LORD: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be my people. And no longer shall each one teach his neighbor and each his brother, saying, 'Know the LORD,' for they shall all know me, from the least of them to the greatest, declares the LORD. For I will forgive their iniquity, and I will remember their sin no more." This time around, God did not write the covenant on stone tablets or scrolls, but he wrote the covenant in the hearts of his people! As a matter of fact, in Ezekiel 36:26-27, God promises, "And I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put my Spirit within you, and cause you to walk in my statutes and be careful to obey my rules."
So, what is this new covenant? This covenant is God providing salvation to people in bondange, changing sinners into saints, and accomplishing his purpose to glorify himself through the Suffering Servant/Man of Sorrows/Messiah/Jesus Christ/King of Kings! "This cup is the new covenant in my blood."