At every Mass, a priest re-enacts the very first Mass that Jesus performed at the Last Supper, and in so doing, consecrates unleavened bread and wine into the flesh and blood of Jesus Himself - the Lamb of God.
Receiving Holy Communion therefore is the Passover Meal of the New Covenant - perfected from the old covenant Jewish Passover meal, and is the communion of the recipient with Jesus, and the rest of His Mystical Body (the Church), in a physical as well as spiritual union.
For many, this is a difficult belief of Catholics for non-Catholics to accept. The disciples explicitly recognise that people would find this core Catholic faith (dogma) difficult when they said "This is a hard saying; who can listen to it?" (John 6:60), but Jesus was clear when he said "I am the living bread which came down from heaven; if any one eats of this bread, he will live for ever; and the bread which I shall give for the life of the world is my flesh." (John 6:51) and "Truly, truly, I say to you, unless you eat the flesh of the Son of man and drink his blood, you have no life in you; he who eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day. For my flesh is food indeed, and my blood is drink indeed. He who eats my flesh and drinks my blood abides in me, and I in him." (John 6:53-56).
When people left Jesus because of their lack of faith in this teaching of Christ, it was Peter - the first Pope, who said "Lord, to whom shall we go? You have the words of eternal life" (John 6:68), and now, the successor of Peter, Pope Francis offers the same Eucharist to us, that all may believe and have life within them, and that all may eat this new fruit of the tree of the cross and not die.
The reception of Holy Communion is therefore a profound event. It is a union with the God-Man, and it is a union with His Mystical Body - the Catholic Church. Therefore ONLY Catholics without the stain of mortal sin may receive Our Lord in Holy Communion. Please be vigilant for those who dishonestly or ignorantly receive Holy Communion and seek to take away the host and not consume. This is a deeply grave act.
To know and understand more about what Catholics believe, see the Catechism of the Catholic Church which sets it out in detail.