Jesus Christ is our eternal priest according to the order of Melchizedek. As. St. Thomas Aquinas says, the proper task of the priest consists in being a mediator between God and men (S. Th. III, 22, 1.). And St. Paul instructs us in 1 Tim. 2:5 that Jesus is the one mediator between God and men. Because Jesus is both God and man by reason of the Hypostatic Union, he is able to reconcile God and man in his own Person. That is what he did by his death on the Cross.
The Catholic Faith teaches that the priest is another Christ, an alter Christus; he takes on his sublime dignity by receiving the Sacrament of Holy Orders. The Most essential function of the sacredotal or priestly office is sacrifice. For we read in Hebrews 8:3: "Every high priest is appointed to offer gifts and sacrifices." It is for this reason, therefore, that the Catholic Church (and others also) calls its clergy "priests." Most Protestants avoid the title because they don't believe that, what they call "The Lord's Supper," is a sacrifice.
I offer this very brief summary of the Church's teaching on sacrifice, Calvary and the Mass because so many, many Catholics, poorly instructed, simply don't understand what is going on when they attend the Sunday Liturgy. That may also explain why many don't understand the difference between "priest" and "minister." The Catholic priest is also a minister, but he is first and foremost a priest because he has been empowered by Christ to offer the sacrifice of the Mass for the Salvation of souls.
Excerpts from Why Are Catholic Priest Called "Priests"? by kenneth Baker, S.J.,