I borrowed this from my church website. I thought it was a nice summary explanation of what is going on in the Catholic liturgy of the Eucharist (and I suspect Orthodox, but I don't want to be presumptuous).

I know that some our christian brothers and sisters don't share this understanding of the Eucharist, but I am sharing this just as a point of reference to maybe shed some light and spread some understanding.

There are three key themes that are woven throughout the Eucharistic prayers. Understanding them can serve to guide us into their deeper meaning. They are not progressive, building off one another, but are woven throughout the prayers like a golden thread tying them together.

1. Anamnesis
Anamnesis is a Greek word that means "remembrance" or "calling to mind." This is more than a simple intellectual remembrance, like remembering where you left the car keys. It's a remembrance that tells us who we are and gives direction to our lives. The opposite of anamnesis-remembrance isn't forgetfulness, but amnesia. Without anamnesis we have no clue where we came from, what we're doing here, or where we're headed. But with anamnesis we recognize that everything comes from God, our life is a gift from God, and we are heading back to God. This truth is beautifully proclaimed in Eucharistic Prayer IV:

"[Y]ou are the one God living and true,
Existing before all ages and abiding for all eternity,
Dwelling in unapproachable light;
Yet you, who alone are good, the source of life,
Have made all that is,
So that you might fill your creatures with blessings
And bring joy to many of them by the glory of your light."

We come from God, our lives are filled with God's blessings, and we are headed back to God. But we aren't content to remember what God did for us just in creating us. We also call to mind what He has done for us to save us the Incarnation, passion, death, Resurrection, and Ascension of Jesus. As we pray in Eucharistic Prayer I:

"O Lord
We celebrate the memorial of the blessed Passion,
The Resurrection from the dead,
And the glorious Ascension into heaven
Of Christ, your Son, our Lord"

When we call to mind all the good things that God has done for us creating us, filling us with His blessings, saving us, and calling us to live eternally in His love, we are pulled automatically into the next great theme of the Eucharistic Prayers

2. Eucharistia
That word should look familiar. What's the first thing you think of when you see eucharistia? Eucharist. And what's the first thing you think of when you think of "Eucharist?" A little white host on the altar? Holy Communion? The truth is, those things are only one part of the meaning of eucharistia.

The Greek word eucharistia means "giving of thanks." When we really call to mind ( anamnesis) when we really consider what God has done for us by creating us and redeeming us, the only natural response is to give thanks to Him to eucharistia Him for his mercy and kindness. But this isn't the empty "thanks" you gave when you got those much-needed (but, let's be honest, much-dreaded) new socks for Christmas. It's more along the lines of the "thank you" of Franciszek Gajowniczek, the man St. Maximilian Kolbe saved from execution in Auschwitz by offering himself up instead. To express his thanks "for the gift of life" Gajowniczek declared, "So long as I have breath in my lungs, I will consider it my duty to tell people about the heroic act of love by Maximilian Kolbe." This kind of eucharistia-gratitude should be so intense that we happily dedicate our lives to God, giving ourselves back to the One who gave Himself up for us. And that's exactly the attitude we see illustrated in Eucharistic Prayer III:

"May he make of us
An eternal offering to you" (EP III)

But we have an issue here. God won't be outdone in generosity. So while He certainly receives the gift that we make of ourselves back to Him in eucharistia-thanksgiving for all that He has done for us, He has more still to give us

3. Koinonia
The Greek word koinonia has many different translations, but for our purposes here we will focus on its translation as "communion" and "fellowship."

In offering ourselves in gratitude to God through the bread and wine placed on the altar, God receives our gifts. Then He does something truly amazing with them through the ministry of the priest. He transforms them into the Body and Blood of Christ, then the God of the Universe , who is love itself, offers it back to us! When we partake of the Body and Blood of Christ in Holy Communion ( koinonia), not only are we united to Christ as individuals. We are also united to one another through Christ's Body and Blood! By sharing in the Body and Blood of Christ, we become the koinonia-fellowship of His Mystical Body on earth! This is God's response to St. John the Baptist's statement that "I must decrease and He must increase." And the Eucharistic Prayers beautifully illustrate:

"Humbly we pray
that, partaking of the Body and Blood of Christ,
we may be gathered into one by the Holy Spirit." (EP II)

"[G]rant that we, who are nourished
By the Body and Blood of your Son
And filled with his Holy Spirit,
May become one body, one spirit in Christ." (EP III)

In Holy Communion we become one with Christ on a deeply personal and individual level. But salvation is never a completely individual matter. We are saved as members of the community ( koinonia) that makes up the Body of Christ on earth through koinonia-communion in His Body and Blood.