What are the links between the story of the Last Supper and the Holy Communion service?

Roman Catholic Answer

A "Holy Communion" service is either when there is no priest available and Communion is distributed to the faithful in absence of a priest OR you are talking about a Protestant service, in which case, you need to ask over in that category.

If you are referring to the Holy Eucharist or the Mass then it has everything to do with the Last Supper. The Last Supper was the Institution of the Eucharist which was a foreshadowing of Jesus' sacrifice on the Cross the following day. The Holy Eucharist is a re-presentation of that same sacrifice.

Look at it this way. God exists, He just is, He can not change, for Him there is no yesterday or tomorrow, as the Bible puts it, "a thousand years are as a watch in the night". From God's point of view He created the world. Put your hands out in front of you on the edge of the desk, as far as you can spread them. Now pretend your left hand is the beginning of the world, and your right hand is the end of the world. You are God, you look down and, BAM, create the world-from Your point of view (God's) He created the beginning, middle, and end all at the same time. You and I are just creatures WHO CHANGE. We live on that line, so we can only see what is right in front of us, or right behind us.

The Last Supper and the Holy Eucharist (the Mass) are both looking at Jesus' crucifixion from God's point of view. When you go into a Catholic church to attend Mass, what is happening on that altar is NOT a reenactment, not just a memorial, NOT just a representation, it is the actual Sacrifice of Our Blessed Lord on the Cross. He is NOT dying again, you are witnessing the actual one time event from God's point of view.

Now, the Last Supper was looking forward to the Crucifixion. On our "time line" in my previous example, they could not see it yet with their eyes, but they could with the eyes of faith. Jesus was making His Sacrifice and His Grace, through His Body and Blood available to His apostles right then.

The Holy Mass is the same Last Supper, we are not doing it again, we are NOT crucifying Jesus again, God is presenting, through His priest, Jesus sacrifice on Calvary in 30 A.D. It is just as if you were standing there on Calvary that dreadful afternoon with the apostle, John, and the Blessed Virgin Mary.
The actual form of it: the Last Supper was the form of the Passover supper:
from the Catechism:

1340 By celebrating the Last Supper in the course of the Passover meal, Jesus gave the Jewish Passover its definitive meaning. Jesus' passing over to his father by his death and Resurrection, the new Passover, is anticipated in the Supper and celebrated in the Eucharist, which fulfills the Jewish Passover and anticipates the final Passover of the Church in the glory of the kingdom.

The "cup" from the Last Supper was the third of the four cups that are used in the Passover, when Jesus said He would not drink wine again until He came into the kingdom, He interrupted the Jewish Passover, and didn't end it until He drank the wine on the Cross - the fourth cup, and culmination of God's Passover.

The present form of the Eucharist that you can see in any Catholic church is two parts: the Liturgy of the Word (the first half), and the Liturgy of the Eucharist (the second half). The first half is from the Passover dinner with the readings, etc. The second is the actual offering of Christ on the cross and then for our participation in His sacrifice through Holy Communion.