(11/26 edit: I left out perhaps the most obvious meaning of "Communion", which I have now included below.)
The Catholic Church, unlike many other Christian communities, has a discipline known as "closed communion." This means that (other than very exceptional cases), only members of the Catholic Church may partake of Holy Communion (that is, receive the Holy Eucharist). Some non-Catholics are very confused by this policy. They see it as exclusive: "Why aren't all invited to the table or the Lord? Did Jesus exclude anyone?" They see it as divisive between Christians: "We allow non-[XYZ]'s to receive the Lord's Supper, so why don't Catholics?" Some are deeply hurt by not being able to receive Communion.
To explain this discipline, we must describe accurately what the Eucharist is and what Holy Communion means to the Church.
Regardless of what a person thinks the Eucharist is, no matter what a person thinks he is eating when he receives Holy Communion in a Catholic church, he is receiving the Body, Blood, Soul, and Divinity of our Lord and Savior. He is not receiving bread or wine, nor a "symbol". If this blessed gift is received unworthily, he adds sin (unworthy reception) to sin (which made his reception unworthy in the first place). Furthermore, this is not just a "meal" or "banquet." This is the Marriage Supper of the Lamb under the veil of a sacrament; this is the consumption of a sacrificial offering.
This Marriage Supper, this mystical wedding banquet, is for those who are "wedded" to Christ in His Church. In other words, receiving Holy Communion means that you are in communion with Christ and His Church. That naturally excludes those who are not Christian at all, the unbaptized. Just as St. Paul wrote that the "fathers" of Israel "all were baptized into Moses in the cloud and in the sea, and all ate the same supernatural food and all drank the same supernatural drink" (1 Cor 10:2-4), so those who are baptized into Christ receive a supernatural food and drink. It was only the Israelites (those of the older covenant) who partook of that supernatural food and drink; likewise, it is only the Church (those of the new covenant, not just one nation) who partake of this supernatural food and drink.
It also excludes those who are Christians but are separated (or "estranged," you could say) from the Bride of Christ, which is the Catholic Church, whether by mortal sin or by not holding the Catholic faith. Faith is a matter of fidelity to God; the Church is faithful to her spouse, Christ, and so her members too must be faithful, believing that Catholic faith.
Catholic doctrine is that the Eucharist is a true sacrifice offered to God. (Numerically, it is the same sacrifice as that of Christ on the cross, with only the manner of the offering being different: the Eucharist is unbloody. Likewise, the Body which is received in the sacrament is numerically identical to that which was conceived in the womb of the Blessed Virgin Mary and died on Calvary.) Let me repeat: the Eucharist is a sacrifice; receiving Holy Communion is partaking in a sacrificial meal. As St. Paul asked, "Do you not know that those who are employed in the temple service get their food from the temple, and those who serve at the altar share in the sacrificial offerings?" (1 Cor 9:13) "Consider the people of Israel; are not those who eat the sacrifices partners in the altar?" (1 Cor 10:18) You are partaking in a sacrificial offering at an altar.
If, as some Protestants believe, the Mass is a satanic and pagan corruption of true worship of God, and the Eucharist is a satanic and pagan sacrifice, then the Eucharist would be being offered "to demons and not to God" (1 Cor 10:20), and anyone who receives Holy Communion at Mass is partaking of the "table of demons." (1 Cor 10:21) What did St. Paul say about that? "You cannot drink the cup of the Lord and the cup of demons. You cannot partake of the table of the Lord and the table of demons." (1 Cor 10:21)
Regardless of whether the Catholic doctrine on the Eucharist is true (and of course, I believe that it is), Catholics believe they are offering a sacrifice to God and are then partaking of that offering. If they are right, and you want to receive it too, why aren't you Catholic?! And if they are wrong, and it is a pagan offering and a fellowship with demons, why would you want to receive it?!