In Catholicism, we believe every baptized Christian participates in the common priesthood (Catechism of the Catholic Church 1547). A priest offers sacrifice to God. In that case what will be the sacrifice in common priesthood?
While it's true that it's the role and prerogative of the ordained priest to offer the sacrifice of Calvary on behalf of the people, the Mass is full of instances showing that it is also the people's Sacrifice (in the context of the Mass, the Sacrifice is Christ's once-for-all Sacrifice, mingled together, or rather offered together, for the greatest effect, the sacrifice(s) of believers consciously united with His): what makes the ordained priest unique is the exclusive ability to confect the Eucharist, something laity cannot ever do, in addition to his right as priest to perform the Sacraments (excepting Baptism and Matrimony, only a priest can perform the Sacraments technically speaking).
A representative example of this concept of the 'united effort' of priest as leader, and laity as united with him as their representative head, is found in the old Latin Rite (essentially Mass in the West for most people for most centuries in the history of the Western part of the Church, up to around 60 years ago and the introduction of the new Mass, with slightly different wording): "Pray, brethren, that my sacrifice and yours [ut meum ac vestrum sacrificium] may be acceptible to God the Father Almighty."
In this sense, laity, all believers, are also priests!
Except you perhaps mean outside the context of the Mass (the Sacrifice).
The priesthood of all believers refers to the fact that offering to God in sacrifice is inherently and historically a priestly act (and not only within Christianity). It is in this context that our prayers and personal penances, good works, and anything which can be "offered up" to God, essentially, is viewed as sacrifice.
Hebrews 13:10-16 (DRB) We have an altar, whereof they have no power to eat who serve the tabernacle. 11 For the bodies of those beasts, whose blood is brought into the holies by the high priest for sin, are burned without the camp. 12 Wherefore Jesus also, that he might sanctify the people by his own blood, suffered without the gate. 13 Let us go forth therefore to him without the camp, bearing his reproach. 14 For we have not here a lasting city, but we seek one that is to come. 15 By him therefore let us offer the sacrifice of praise always to God, that is to say, the fruit of lips confessing to his name. 16 And do not forget to do good, and to impart; for by such sacrifices God's favour is obtained.
Ministerial priesthood ≠ "common priesthood" of all the baptized.
Pope Pius XII's encyclical Mediator Dei §83 condemns those who think there is only one priesthood common to all the baptized:
- For there are today, Venerable Brethren, those who, approximating to errors long since condemned* teach that in the New Testament by the word "priesthood" is meant only that priesthood which applies to all who have been baptized; and hold that the command by which Christ gave power to His apostles at the Last Supper to do what He Himself had done, applies directly to the entire Christian Church, and that thence, and thence only, arises the hierarchical priesthood. Hence they assert that the people are possessed of a true priestly power, while the priest only acts in virtue of an office committed to him by the community. Wherefore, they look on the eucharistic sacrifice as a "concelebration," in the literal meaning of that term, and consider it more fitting that priests should "concelebrate" with the people present than that they should offer the sacrifice privately when the people are absent.
- […] Hence, he [the priest] goes to the altar as the minister of Christ, inferior to Christ but superior to the people.† The people, on the other hand, since they in no sense represent the divine Redeemer and are not mediator between themselves and God, can in no way possess the sacerdotal power (iure sacerdotali).
- All this has the certitude of faith. However, it must also be said that the faithful do offer the divine Victim, though in a different sense (diversa ratione).
*Cf. Council of Trent, Sess. 23. c. 4.
†Cf. Saint Robert Bellarmine, De Missa, 2, c.4.
Thus, only ordained priests, by perpetuating Christ's sacrifice throughout the ages in the Holy Mass, offer Himself to God the Father.
cf. ch. 6 (PDF pp. 23-31) of Vatican II preparatory commission's De Ecclesia
#Laity's sacrifices are united to the priest's sacrifice of Our Lord.
Pope Pius XI's encyclical Miserentissimus Redemptor explains that laypeople are to offer sacrifice "in much the same manner as every [ordained, ministerial] priest":
- […] Nor do those [i.e., ministerial priests] only enjoy a participation in this mystic priesthood and in the office of satisfying and sacrificing, whom our Pontiff Christ Jesus uses as His ministers to offer up the clean oblation to God's Name in every place from the rising of the sun to the going down (Malachias i, 11), but the whole Christian people rightly called by the Prince of the Apostles "a chosen generation, a kingly priesthood" (1 Peter ii, 9), ought to offer for sins both for itself and for all mankind (Cf. Hebrews v, 3), in much the same manner [haud aliter propemodum] as every priest and pontiff "taken from among men, is ordained for men in the things that appertain to God" (Hebrews v, 1).
He continues by explaining how all the baptized's sacrifices are united to the priest's:
- But the more perfectly that our oblation and sacrifice corresponds to the sacrifice of Our Lord, that is to say, the more perfectly we have immolated our love and our desires and have crucified our flesh by that mystic crucifixion of which the Apostle speaks, the more abundant fruits of that propitiation and expiation shall we receive for ourselves and for others. For there is a wondrous and close union of all the faithful with Christ, such as that which prevails between the head and the other members[…]
See also part 2, ch. 31, § "How to Unite Ourselves to the Eucharistic Sacrifice" of The Three Ages of the Interior Life: Prelude of Eternal Life by Fr. Réginald Garrigou-Lagrange, O.P.
We don't offer our sacrifice solely in that sense. We offer the sacrifice of Jesus.
The sacrifice of Jesus "for the sins of the whole world"416 expresses his loving communion with the Father. "The Father loves me, because I lay down my life", said the Lord, "[for] I do as the Father has commanded me, so that the world may know that I love the Father. C606
For example: in divine mercy chaplet we pray:
Eternal Father, I offer you the Body and Blood, Soul and Divinity of Your Dearly Beloved Son, Our Lord, Jesus Christ, in atonement for our sins and those of the whole world.
and with this prayer we can say: in unity with my sacrifice(fasting, this prayer, etc.) through Mary's unblemished hands and the Divine Heart of Jesus I offer this for here-say-your-intention...
Here is also a prayer similar to the Chaplet of divine mercy.