The Catechism of the Catholic Church tells us that the word "catholic" means "universal." The word “universal” is understood in the sense of "in keeping with the whole." The Church is catholic in a double sense: First, the Church is catholic because Christ is present in her. Where there is Christ Jesus, there is the Catholic Church. In her subsists the fullness of Christ's body united with its head; this implies that she receives from Christ Himself, the fullness of the means of salvation. The Church believes that Christ has willed and declared the “means of our salvation” to mean: correct and complete confession of faith, full sacramental life, and ordained ministry in apostolic succession. The Church was, in this fundamental sense, catholic on the day of Pentecost and will always be so until the day of Christ’s return.
Secondly, the Church is catholic because Christ has sent her out on a mission to the whole of the human race. All are called to belong to the new People of God. This People, therefore, while remaining one and only one, is to be spread throughout the whole world and to all ages in order that the design of God's will may be fulfilled. He made human nature one in the beginning and has decreed that all his children, who were scattered, should be finally gathered together as one. The character of universality, which adorns the People of God, is a gift from the Lord himself whereby the Catholic Church ceaselessly and efficaciously seeks for the return of all humanity and all its goods, under Christ the Head in the unity of his Spirit.
Each particular Church is "catholic"
The Church of Christ is really present in all legitimately organized local groups of the faithful, which, in so far as they are united to their pastors, are also quite appropriately called Churches in the New Testament. In these Churches, the faithful are gathered together through the preaching of the Gospel of Christ, and the mystery of the Lord's Supper is celebrated. In these communities, regardless of where they exist, whether they are small or large, rich or poor, Christ is present, through whose power and influence the One, Holy, Catholic, and Apostolic Church is constituted. The phrase "particular church," which is first of all the diocese, refers to a Catholic community of the Christian faithful, in communion of faith and sacraments with their bishop, ordained in apostolic succession. These particular Churches are constituted after the model of the universal Church. It is in these particular Churches, and formed from these particular Churches, that the one and unique Catholic Church exists.
Particular Churches are fully catholic through their communion with, the Church of Rome, which presides in charity. For with this church, by reason of its pre-eminence, the whole Church, that is the faithful everywhere, must necessarily be in solidarity. Indeed, from the incarnate Word's descent to us, all Christian churches everywhere have held and hold the great Church in Rome to be their only basis and foundation since, according to the Savior's promise, the gates of hell have never prevailed against her.
We must be very careful not to conceive of the universal Church as the simple sum, or federation, of essentially different particular churches. In the mind of the Lord, the Church is universal by vocation and mission, but when she puts down her roots in a variety of cultural, social, and human terrains, she takes on different external expressions and appearances in each part of the world. The rich variety of ecclesiastical disciplines, liturgical rites, and theological and spiritual heritages, proper to the local churches, unified in a common effort, shows all the more resplendently the catholicity of the undivided Church.
Who belongs to the Catholic Church?
All are called to this catholic unity of the People of God. And to it, in different ways, belong the Catholic faithful, others who believe in Christ, and finally all Humanity, called by God's grace to salvation. Fully incorporated into the society of the Church are those who, possessing the Spirit of Christ, and accept all the means of salvation given to the Church. This includes the Churches entire organization. Further, those who, by the bonds constituted by the profession of faith, the sacraments, ecclesiastical government, and communion, are joined in the visible structure of the Church of Christ, who rules her through the Supreme Pontiff and the bishops. In terms of personal salvation, it is not enough to be incorporated into the Church, one must also persevere in charity. Those who remain in the bosom of the Church, live and work as part of her body, believe and understand with her mind, act with her spirit and love with her heart.
The Church knows that she is joined, in many ways, to the baptized, who are honored by the name of Christian, but do not profess the Catholic faith in its entirety, or have not preserved unity or communion under the successor of Peter. Those who believe in Christ, and have been properly baptized, share in an imperfect communion with the Catholic Church. With the Orthodox Churches, this communion is so profound that it lacks little to attain the fullness that would permit a common celebration of the Lord's Eucharist.
(Catechism of the Catholic Church - 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 771, 795, 811, 814, 815, 815, 816, 818, 830, 831, 831, 832, 833, 834, 835, 836, 837, 838, 882, 882, 886, 1202, 1271, 1369)
Summarized by Rev. E. C. Tozzi
Catechism of the Catholic Church. Washington, DC: United States Catholic Conference, 1994.