And on the first day of the Azymes, the disciples came to Jesus, saying: Where wilt thou that we prepare for thee to eat the pasch?  But Jesus said: Go ye into the city to a certain man, and say to him: the master saith, My time is near at hand, with thee I make the pasch with my disciples.  And the disciples did as Jesus appointed to them, and they prepared the pasch.  But when it was evening, he sat down with his twelve disciples.
 And he said to them: With desire I have desired to eat this pasch with you, before I suffer.
 For I say to you, that from this time I will not eat it, till it be fulfilled in the kingdom of God.
Praised be Jesus Christ!
Now and forever. Amen.
I'm going to be perfectly honest with you. With respect to this particular Mystery, you will probably have read, or attended talks, or watched YouTube presentations on this subject that go much more deeply into it than I will. For example, you can watch Dr. Scott Hahn's talk on YouTube called The Fourth Cup, which I've linked here.
Given that, I hope you will forgive me if my exploration of this event is less exhaustive, but I'll do what I can.
The first thing I want to say is that this event is the source of the Catholic Church's teaching on the Eucharist, which I will explain briefly here. The Church teaches that, at the Last Supper, when Jesus spoke the words that He did over the bread and wine, that He was instituting what we call the "Eucharist," which comes from the Greek word eukharistia, which means "thanksgiving."
But, while other (but certainly not all) non-Catholic Christian groups believe that the Eucharist is a commemorative meal, and that the bread and wine taken at the meal are symbolic in nature, the Catholic Church teaches differently. Rather, the Church teaches that when a consecrated minister (priest or bishop) speaks the words that Jesus did at the Last Supper, over the bread and wine, that the substance of these things change into the actual, physical, body and blood (along with His soul and Divinity, as His natures are indivisible). The Church calls this change transubstantiation, which indicates a change in the substance only (and not, what is called in philosophy, the "accidents" of the bread and wine, which refers to its material appearance).
In this way, when Jesus says He is the True Bread from Heaven (referring to the Manna that the Israelites ate in the desert), as well as the True Vine, He intends for us to understand that we are to consume Him in the Eucharistic meal, which He institutes as the fulfillment of the Paschal meal, to be perpetuated throughout time.
This is important to understand, because in the Gospels, we see really significant events at every one of the Paschs of Jesus' ministry. We see the transformation of water into wine at the wedding of Cana. And on two other occasions (near the Pasch), Jesus feeds 5,000, and 4,000 people with a handful of bread (and fish, of course). The point here is that during Jesus ministry, we have a miracle of transformation, and two miracles of multiplication during the Pasch, and on His final Pasch before being Crucified, He institutes the Eucharist, which is both transformative (transubstantiation of the bread and wine into His Body and Blood), and multiplicative (He is fully present in all "bits" of the Eucharistic meal that He gives to us).
So, at the wedding feast of Cana, He transforms water into wine, so that the guests of the wedding might have their celebration enriched, and at the Last Supper, He transubstantiates the wine into the True Wine (Himself), that our spiritual lives might be enriched. And, again, He multiplies the bread to feed, on more than one occasion, thousands of people, to meet their need for physical nourishment, and at the Last Supper He multiplies His physical presence into all consecrated hosts that we all may receive the spiritual nourishment that we are in need of.
He Shall Betray Me
Now, the Gospels of Matthew, Mark and Luke each recount the events of the Last Supper, pretty much the same, and you can read that account above. John doesn't record the Eucharistic meal in His Gospel, but instead records Jesus' last discourse with the Apostles before His Passion.
However, even though John's account doesn't focus on the meal, He does include this part of the story. "Jesus answered: He it is to whom I shall reach bread dipped. And when he had dipped the bread, he gave it to Judas Iscariot, the son of Simon." (John 13:26) This is specifically mentioned in all four Gospels. And, if you know anything about the Gospels, you'll know that it is uncommon for all four of them to explicit recount exactly the same thing.
So, this is important.
In typical Protestant theology, there are two doctrines that they teach in contradiction to Catholic teaching. The first is that when Jesus is speaking about eating His body, or eating His flesh, even if He calls it bread, He's really just referring to His teachings. They reference Jesus' teaching on eating the bread of the Pharisees, which He explained was a reference to accepting their doctrines (Matthew 16:6-12). So, to eat the flesh of Jesus, to them means to accept His teachings, and His sacrifice.
The second is that they typically believe (though, of course, neither this teaching nor the one I just mentioned is universal among all denominations, but these are common) that once you are saved, you will always be saved, and that you become saved by accepting Jesus teachings and His sacrifice, as He taught (to eat His flesh and drink His blood).
In my opinion, this occurrence at the Last Supper confounds these two doctrines. Because if eating the bread, which is symbolic of Jesus' doctrines, is a symbolic act of taking in His doctrine within us, then we can see clearly at the Last Supper that Judas Iscariot did so, which, according to typical protestant doctrine, would indicate he was saved. But then He betrays Jesus, almost immediately, confounding the idea that He would always be saved. Especially, since Jesus says of him who betrays Him that it were better he was never born, implying his damnation.
We see a warning about this in 1 Corinthians 11:27-29. And here, St. Paul is speaking specifically about the practice of the Eucharistic meal, not about doctrine. This refutes the idea that Jesus' teaching about eating His flesh and blood is simply about accepting His doctrine, otherwise how could one be "guilty of the body" if it were simply about doctrine. If it were, then one would be "guilty of the bread," which is what Jesus' teaching about the leaven of the Pharisees was about.
But, really, I think this is a dire warning to us, who believe, more than anything. Judas was partaking of the meal, and had already made the decision to betray Jesus. Jesus makes it a point to reveal His betrayer through the meal. Think of the thousands of different ways Jesus could have revealed Judas for who he was, and on hundreds of different occasions. Instead, He chose that moment when He instituted the most intimate communion of persons, between Himself and all of us.
That should strike you as troubling, I should hope, because by it He is showing the importance of this event. Aren't we all Judas Iscariot? Do we not all go to Mass, take communion, then go home and betray Jesus by our sins that we almost never cease doing? This is why the Church has in place two very important restrictions when it comes to partaking of the Eucharist.
First, you must know what it is you do. If you are not Catholic, and do not believe the Eucharist to be what it is, Jesus' True Presence, then you are not permitted to enter into a communion with Him, and His community of believers, when doing so would be a lie.
Second, you must be in a state of Sanctifying Grace. That is, you must not have any unrepentant mortal sins on your soul. For, if you do, then you are no better than the Betrayer, and you partake of the Eucharistic meal, again, in falsehood. You blaspheme the Sacrament. And, as Jesus said, it would be better if you were not born (true of all sins that destroy your soul to death).
A Hymn Being Said
But the Eucharist is a gift! It is a great gift of joy and thankfulness! By it, we are permitted to enter into communion with the Trinity, into their community of Love. We call full communion (of the kind we will be permitted in Heaven) with this community the Beatific Vision because it is beautiful, and joyous and filled with love. The Eucharist is supposed to be, for us, a taste of that vision. We call it the Eucharistic Celebration, or celebration of thanksgiving for the great gift of Jesus that we have received.
This is why the Eucharist is called by the Church the source and summit of our Faith (Catechism of the Catholic Church 1324). It is called the source of our Faith because it feeds our Faith, gives it life, nourishes it (as bread it satisfies our need). Through communion with Jesus, who is God, the source of all life and love, our spiritual lives are sustained, and given rebirth each time we partake of this food. It is called the summit of our Faith because it enriches our Faith, makes it sweet and lovely (as wine, it enhances our spiritual experience with beauty). Through communion with Jesus, who is God, the source of all goodness and beauty, our spirits are uplifted to heights unthinkable by our own efforts.
And so, in joy and love, the Apostles, with Jesus, sang a Paschal hymn. And with that, I'm going to leave you with a beautiful old roman Catholic chant.