From the CCC:
III. THE CHURCH IS CATHOLIC
What does “catholic” mean?
830 The word “catholic” means “universal,” in the sense of “according to the totality” or “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."307 In her subsists the fullness of Christ’s body united with its head; this implies that she receives from him "the fullness of the means of salvation"308 which he has willed: 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 Pentecost309 and will always be so until the day of the Parousia.
831 Secondly, the Church is catholic because she has been sent out by Christ on a mission to the whole of the human race:310
All men 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.311
Each particular Church is “catholic”
832 "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 them 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, though they may often be small and poor, or existing in the diaspora, Christ is present, through whose power and influence the One, Holy, Catholic, and Apostolic Church is constituted."312
833 The phrase “particular Church,” which is first of all the diocese (or eparchy), refers to a community of the Christian faithful in communion of faith and sacraments with their bishop ordained in apostolic succession.313 These particular Churches "are constituted after the model of the universal Church; it is in these and formed out of them that the one and unique Catholic Church exists."314
834 Particular Churches are fully catholic through their communion with one of them, the Church of Rome "which presides in charity."315 "For with this church, by reason of its pre-eminence, the whole Church, that is the faithful everywhere, must necessarily be in accord."316 Indeed, "from the incarnate Word’s descent to us, all Christian churches everywhere have held and hold the great Church that is here [at Rome] to be their only basis and foundation since, according to the Savior’s promise, the gates of hell have never prevailed against her."317
835 "Let us be very careful not to conceive of the universal Church as the simple sum, or . . . the more or less anomalous federation of essentially different particular churches. In the mind of the Lord the Church is universal by vocation and mission, but when she put 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."318 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."319
Who belongs to the Catholic Church?
836 "All men are called to this catholic unity of the People of God. . . . And to it, in different ways, belong or are ordered: the Catholic faithful, others who believe in Christ, and finally all mankind, called by God’s grace to salvation."320
837 "Fully incorporated into the society of the Church are those who, possessing the Spirit of Christ, accept all the means of salvation given to the Church together with her entire organization, and 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. Even though incorporated into the Church, one who does not however persevere in charity is not saved. He remains indeed in the bosom of the Church, but ‘in body’ not ‘in heart.’"321