At weekday Masses in the Catholic Answers chapel, Communion is by intinction and thus under both kinds. On Sundays, at my parish, I receive only the host and not from the cup, which is to say I receive under one kind only. If someone were to ask me why, and if I felt mischievous, I might reply, "Because I’m not an Utraquist, of course!" My questioner would respond, "Huh?" Then I’d give him a short lesson in history and sacramental theology.
In 1414, Jacob of Mies, professor of philosophy at the University of Prague, began to teach a novel theory. He said that, to be saved, a Catholic must receive Communion under both kinds and that laymen were being shortchanged because they were offered only the host. The chalice was reserved for the priest alone. Jacob’s theory was picked up by the followers of Jan Hus, who had been excommunicated in 1411 and who ended up being burned at the stake in 1415. Hussites became famous for insisting on the laity having access to the Communion cup.
Jacob came to his new teaching by reading John 6:53: "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." To him this meant that to receive only the host or only from the cup meant a partial Communion and thus no salvation. Under the form of bread, he thought, one received only Christ’s body and under the form of wine only his blood. To receive the whole Christ, and to be obedient to the biblical injunction, one had to receive under both kinds (sub utraque specie, therefore "Utraquist" as the name for those who hold this position).
Jacob’s teaching was condemned by the councils of Constance, Basil, and Trent. Not only did it contradict long-established usage, but it also contradicted Scripture: "Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of profaning the body and blood of the Lord" (1 Cor. 11:27).
Pay attention to the conjunctions here. Paul says that if someone receives either the host or from the cup and does so when not in the state of sanctifying grace ("in an unworthy manner"), he profanes both, the body and the blood of the Lord. This implies that the body and the blood are equally present under the form of bread and under the form of wine. Either species contains the whole Christ. If you receive Christ at all, you receive all of him.
Not all Catholics understood this in the fifteenth century, and not all understand it today, which is why there are functional Utraquists in our parishes. Their confusion reminds me of a related one that was manifested at a local parish that is frequented by tourists. One particularly crowded Sunday, the priest ran short of hosts and had to break the last few dozen in half to make sure everyone could communicate. Knowing how poorly catechized many Catholics are, at the end of Mass he said, "Don’t worry! Even if you got only half a host, you got the whole Christ."