The Catechism of the Catholic Church describes the common priesthood of the baptized and the priesthood stemming from Holy Orders in the following manner (the below references to a "royal priesthood" are from 1Peter 2:4-5):
1546: Christ, high priest and unique mediator, has made of the Church "a kingdom, priests for his God and Father." The whole community of believers is, as such, priestly. The faithful exercise their baptismal priesthood through their participation, each according to his own vocation, in Christ's mission as priest, prophet, and king. Through the sacraments of Baptism and Confirmation the faithful are "consecrated to be . . . a holy priesthood."
1547: The ministerial or hierarchical priesthood of bishops and priests, and the common priesthood of all the faithful participate, "each in its own proper way, in the one priesthood of Christ." While being "ordered one to another," they differ essentially. In what sense? While the common priesthood of the faithful is exercised by the unfolding of baptismal grace --a life of faith, hope, and charity, a life according to the Spirit--, the ministerial priesthood is at the service of the common priesthood. It is directed at the unfolding of the baptismal grace of all Christians. The ministerial priesthood is a means by which Christ unceasingly builds up and leads his Church. For this reason it is transmitted by its own sacrament, the sacrament of Holy Orders.
An example of the distinction between these two priesthoods can be seen in how one participates in the Mass and other liturgies:
1141: The celebrating assembly is the community of the baptized who, "by regeneration and the anointing of the Holy Spirit, are consecrated to be a spiritual house and a holy priesthood, that through all the works of Christian men they may offer spiritual sacrifices." This "common priesthood" is that of Christ the sole priest, in which all his members participate:
Mother Church earnestly desires that all the faithful should be led to that full, conscious, and active participation in liturgical celebrations which is demanded by the very nature of the liturgy, and to which the Christian people, "a chosen race, a royal priesthood, a holy nation, a redeemed people," have a right and an obligation by reason of their Baptism.
1142: But "the members do not all have the same function." Certain members are called by God, in and through the Church, to a special service of the community. These servants are chosen and consecrated by the sacrament of Holy Orders, by which the Holy Spirit enables them to act in the person of Christ the head, for the service of all the members of the Church. The ordained minister is, as it were, an "icon" of Christ the priest. Since it is in the Eucharist that the sacrament of the Church is made fully visible, it is in his presiding at the Eucharist that the bishop's ministry is most evident, as well as, in communion with him, the ministry of priests and deacons.
The role of the universal priesthood is lived in accordance with the grace that God bestows through baptism. See CCC # 1262 - 1274.
For more information on this topic, please read the following article by Tim Staples: The Priesthood is Both Ministerial and Universal