I wasn’t quite sure, so I looked it up in the Catechism of the Catholic Church. Searching for all occurrences of the phrase “Body of Christ,” I see that it almost always refers to the Church.
That is Church with a capital C, meaning it is not merely an earthly organization of people who love each other and do good works. It is both earthly and divine, The Church is given life by the Holy Spirit. It consists not only of members of the Church who are presently alive, but also those who have died and live eternally in God’s presence, together with Christ (“through Him, with Him, and in Him”).
In several places, the Catechism says we become members of the Body of Christ through Baptism and by the Holy Spirit dwelling in us. This comes perhaps from 1 Corinthians 12:13:
For in the one Spirit we were all baptized into one body.
(though I think Jesus may have said it first, for example, in the Gospel of John).
Not only baptism, but all of the sacraments build up and strengthen the Body of Christ.
The Catechism does not say that every human is a member of the Body of Christ, but in paragraph 776 it says that God desires it:
… God desires “that the whole human race may become one People of God, form one Body of Christ, and be built up into one temple of the Holy Spirit.”
Those who do not follow Christ (are not joined to Christ) are not members of the Body of Christ, though someday they may be.
Another essential point is that the members of the Body of Christ serve each other, collaborating with God in the salvation of others and in the growth of the Body of Christ, the Church.
There is much more. Let me cut and paste several more passages from the Catechism:
II. The Church—Body of Christ
The Church is communion with Jesus
787 From the beginning, Jesus associated his disciples with his own life, revealed the mystery of the Kingdom to them, and gave them a share in his mission, joy, and sufferings. Jesus spoke of a still more intimate communion between him and those who would follow him: “Abide in me, and I in you… I am the vine, you are the branches.” And he proclaimed a mysterious and real communion between his own body and ours: “He who eats my flesh and drinks my blood abides in me, and I in him.”
788 When his visible presence was taken from them, Jesus did not leave his disciples orphans. He promised to remain with them until the end of time; he sent them his Spirit. As a result communion with Jesus has become, in a way, more intense: “By communicating his Spirit, Christ mystically constitutes as his body those brothers of his who are called together from every nation.”
789 The comparison of the Church with the body casts light on the intimate bond between Christ and his Church. Not only is she gathered around him; she is united in him, in his body. Three aspects of the Church as the Body of Christ are to be more specifically noted: the unity of all her members with each other as a result of their union with Christ; Christ as head of the Body; and the Church as bride of Christ.
III. The Church Is the Temple of the Holy Spirit
797 “What the soul is to the human body, the Holy Spirit is to the Body of Christ, which is the Church.” “To this Spirit of Christ, as an invisible principle, is to be ascribed the fact that all the parts of the body are joined one with the other and with their exalted head; for the whole Spirit of Christ is in the head, the whole Spirit is in the body, and the whole Spirit is in each of the members.” The Holy Spirit makes the Church “the temple of the living God”:
Indeed, it is to the Church herself that the “Gift of God” has been entrusted… It is in her that communion with Christ has been deposited, that is to say: the Holy Spirit, the pledge of incorruptibility, the strengthening of our faith and the ladder of our ascent to God… For where the Church is, there also is God’s Spirit; where God’s Spirit is, there is the Church and every grace.
798 The Holy Spirit is “the principle of every vital and truly saving action in each part of the Body.” He works in many ways to build up the whole Body in charity: by God’s Word “which is able to build you up”; by Baptism, through which he forms Christ’s Body; by the sacraments, which give growth and healing to Christ’s members; by “the grace of the apostles, which holds first place among his gifts”; by the virtues, which make us act according to what is good; finally, by the many special graces (called “charisms”), by which he makes the faithful “fit and ready to undertake various tasks and offices for the renewal and building up of the Church.”
953 Communion in charity. In the sanctorum communio, “None of us lives to himself, and none of us dies to himself.” “If one member suffers, all suffer together; if one member is honored, all rejoice together. Now you are the body of Christ and individually members of it.” “Charity does not insist on its own way.” In this solidarity with all men, living or dead, which is founded on the communion of saints, the least of our acts done in charity redounds to the profit of all. Every sin harms this communion.
2003 Grace is first and foremost the gift of the Spirit who justifies and sanctifies us. But grace also includes the gifts that the Spirit grants us to associate us with his work, to enable us to collaborate in the salvation of others and in the growth of the Body of Christ, the Church.