A covenant is a contract or agreement between two parties. According to tradition, a covenant is made between two people when blood is spilt and a life is sacrificed. The person that makes the sacrifice is essentially saying to the other, "If I don't hold up my end of the contract, may what's done to this sacrifice be done to me also."
Jesus talks about the blood of the covenant during the Passover meal with his disciples. The covenant is for forgiveness. We're used to thinking about God sacrificing Jesus for us, God doing this thing to Jesus for us, God willing Jesus to die for us. The thing is, Jesus tells the disciples they have to drink from the cup. That means the disciples have to be the ones to make the sacrifice, not God.
Yes, God sent Jesus, but people killed him.
So, when the disciples drink of the cup that represents the blood of the covenant, they are essentially saying to God, "If I don't hold up my end of the contract, may what's done to this sacrifice be done to me also." In this case, their end of the contract is allowing their beloved teacher to be killed.
Then, Jesus is praying in Gethsemane and he says this funny thing:
"My father, if this cup cannot be taken away from me unless I drink it, your will must be done."
Jesus had a covenant to make with God, too. In order for death to pass from him, he had to accept death. And there's something interesting about the word drink..."to receive into the soul what serves to refresh, strengthen, and nourish into eternal life."
Jesus and the disciples had to receive his death into their souls.
But Jesus' death isn't just about him; it's also about what the disciples had to give up:
They're preconceived notions of a messiah,
Their misled belief that he would take the throne and overthrow Rome,
Their hope that Jesus would deploy a legion of angels upon their enemies,
Their own ideas of what it was to be in right standing with God.
When the disciples drank the cup, they were agreeing to kill the idol that they wanted Jesus to be. And when Jesus drank the cup, he was agreeing to be the idol that needed to be sacrificed.