In the celebration of Holy Communion, the Chalice is the cup in which the wine is consecrated. It is typically made of silver or gold and is often has a decorated base. The beauty of the chalice, however, is not in its external adornment but rather in the very blood of Christ it holds when the wine is consecrated. Jesus, who is both the priest and the sacrifice, offers his own blood, shed for the forgiveness of sins, for all who would worthily receive it by trusting in Christ's words concerning this cup.
Modern practices have allowed for individual cups to be used as an alternative to the Chalice. Despite the theological, historical, and social benefits of those united in Christ to share in the One Cup (1 Corinthians 10:16-17), the individual cup is permitted to ease consciences of those fearful of germs. It should be noted, however, that a study has been done by the Center for Disease Control (CDC) that shows those fears are more perception than reality. An article from the American Journal of Infection Control can be found here.
The Paten, which is Latin and Greek for "plate", is the vessel upon which the bread is consecrated. It serves to carry Christ's body to the people that they may receive it. (Luke 22:19)
The Paten is designed to fit over the chalice prior to and after celebrating the Holy Supper.
"Pyx" means "box" in Greek. Sometimes called the "bread box", the pyx is a small container, typically made of silver or gold, that holds the communion wafers prior to the consecration and distribution of the host. In some circumstances the pyx might be used to carry the body of Christ for distribution. An alternative to the pyx is the ceborium.
Shaped similar to the chalice but with the addition of a cover, the ciborium holds the host in a way that combines the use of the paten and the pyx. When a ciborium is used it replaces the pyx or even the pyx and the paten. In medieval times the ciborium remained on the altar following the celebration of the Eucharist and served as the means of storing the host that remained. The priest would then use it to commune the sick.
The flagon is a gold or silver pitcher used to hold the communion wine. Most congregations do not need the voluminous flagon to celebrate the Lord's Supper, especially since the popularization of the individual cups. A smaller glass container, call a cruet, is more common today.