The cups of wine are drunk at certain times during the Passover meal. The four cups generally represent freedom and redemption. There is joy seen in drinking the wine although there is a prohibition against getting drunk out of fear the Passover Seder wouldn't get finished.
Jesus refers to this cup in Luke 22:17-18.
17 "After taking the cup, He gave thanks and said, "Take this and divide it among you.
18 For I tell you I will not drink again of the fruit of the vine until the kingdom of God comes."
There is probably not a New Testament reference to the second cup.
Jesus refers to this cup in 1 Corinthians 11:25.
1 Cor 11:25 "In the same way, after supper He took the cup, saying, This cup is the new covenant in my blood; do this, whenever you drink it, in remembrance of me. Christians should recognize this liturgy because it is used while taking communion. For more on this please read The Last Supper.
There is probably not a New Testament reference to the fourth cup.
Some believe that the fourth cup, sometimes known as the cup of redemption, will only really be complete when Israel is redeemed by her Messiah. This fourth cup then becomes a sign of hope for the whole world that when Messiah comes there will be peace.
Elijah is a prophetic guest of honor at the Seder meal. A cup of wine is poured for him one time, set on the table, and the door to the house is opened. This traditional act is an invitation for Elijah to join us at our Seder meal. The tradition states that Elijah will show up here on earth to announce that Messiah is coming. This is based on a passage in Malachi.
Malachi 4:5 "See, I will send you the prophet Elijah before that great and dreadful day of the LORD comes."
This understanding that Elijah would usher in the Messianic Era has been understood for at least 2000 years. John the Baptist began to baptize people. (He is considered a type of Elijah). That's why the Pharisees sent Priests and Levite's to investigate. They knew of this prophecy and was expecting the Messiah to come and deliver them from Roman military control. So they asked John "They asked him, “What then? Are you Elijah?” (John 1:21).
For more on this please read about the Connection Between John The Baptist And Elijah.
This cup is also identified with Exodus 6:8, which is interpreted as referring to Israels' final redemption.
This is a great time to sing the song Eliyahu HaNavi.
There are different traditions given for the four cups of wine. No one tradition prevailed. Here is a sample of a couple of them.
The Mishna teaches that according to Rabbi Yohanon and Rabbi Benayah, these four cups correspond to the four verbs in Exodus 6:6-7, describing Gods redemption.
I will bring you out.
I will deliver you.
I will redeem you.
I will take you.
Another one ties in the four cups of wine with the four times the word cup is mentioned by Pharaohs jailed butler as he retells his dream to Joseph while they were in prison together.
There are other Christian symbols that can be seen in the four wine cups. For more on this please read Christ In The Passover.
Article of interest concerning Passover include:
John 2:1-10 talks about the miracle of Messiah turning water into wine.