Often people point to the scriptures that indicate we are all part of the priesthood to disprove that the Catholic Church is the Church of Christ. Catholics certainly believe that all Christians share in the common priesthood, but we believe in addition to that there is a ministerial priesthood as well. It's in the catechism, this quote explains:
The Catechism of the Catholic Church says clearly that all of God’s faithful people share in the priesthood of Christ by virtue of their baptism: "Christ, high priest and unique mediator, has made of the Church ‘a kingdom, priests for his God and Father’" (CCC 1546, cf. Rev. 1:6, 5:9–10). Further, it says, "The whole Church is a priestly people. Through baptism all the faithful share in the priesthood of Christ. This participation is called the ‘common priesthood of all the faithful.’ Based on this common priesthood and ordered to its service, there exists another participation in the mission of Christ: the ministry conferred by the sacrament of holy orders, where the task is to serve in the name and in the person of Christ the head in the midst of the community" (1591). In other words, the existence of a common priesthood for all God’s people does not exclude a special calling for the pastors of the Church to be priests.
This explains the parallel between the priesthood of the Old Testament, and the priesthood of the New Testament:
What we see in the Old Testament is a three-fold priesthood. There is the common or universal priesthood of all Israelites at the bottom (cf. Ex. 19:6), a ministerial priesthood above them (cf. Ex. 19:22, 24; Lev. 1:5), and a high priest at the top (cf. Num. 35: 25). We thus should expect to find a similar three-fold priesthood under the New Covenant, and we do. There is the common or universal priesthood of all Christians (cf. 1 Pet. 2:5, 9), a ministerial priesthood above them (cf. Rom. 15:16), and a high priest at the top (cf. Heb. 3:1). Rather than varying from the biblical model of priesthood, the Catholic understanding copies it exactly.
(Both quotes are from here. )
For more detailed info about the priesthood.