The Church is the congregation of baptized Christians under the authority of a bishop in union with the Bishop of Rome, or Pope (CCC 816). Jesus founded this one church, and that church is the Catholic Church (CCC 763). Even though the Church is visible (members are identifiable by virtue of their corporate membership in the Catholic Church), non-Catholic Christians who have been validly baptized and are in the state of grace also belong to the Catholic Church (CCC 838), although invisibly, and therefore, imperfectly.
The Bible says, "You are Christ's body, and individually members of it" (1 Cor. 12:27). The Church is also called the "Mystical Body of Christ", with Jesus as head (Col. 1:18). The Church is therefore a living entity.
Traditionally, the Church has identified four marks by which we know the Catholic Church is the church that Christ founded and continues to be with. The Creed identifies these marks as being "one, holy, catholic, and apostolic" (CCC 811). The Church is one because it is united under one set of teachings, the same sacraments, and one head–the Pope (CCC 815). The Church is holy because its founder, Jesus Christ, was holy; its teachings and sacraments are holy; and the teachings and sacraments of the Church produce holiness in its members (CCC 823-824). The Church is catholic (universal) because it was founded for all people in all nations of all races (CCC 831). And the Church is apostolic because it was founded on the Apostles, and the bishops that govern the Church are the successors of the Apostles (CCC 857).
When Christ founded His church, the Apostles were given the authority to teach, sanctify, and rule the Church through the power of the Holy Spirit (CCC 857). This authority was passed on to their successors, the bishops, through Holy Orders (CCC 1558). The bishops passed this authority on to other bishops through the laying on of hands (2 Tim. 2:2), and each bishop in the Catholic Church today is in direct succession from one of the Apostles (CCC 1555).
These bishops form what is called the "Magisterium" (or teaching office) of the Catholic Church. They have the authority, passed down from the Apostles themselves, to teach and also to bind the faithful to assent to the doctrines as definitively declared by the Magisterium (CCC 892). They also have the power to make church laws and disciplinary rulings (Matt. 18:18), and bind the Church to them, because obedience to the bishops is obedience to Christ (Luke 10:16). Many of these rules are found in the Code of Canon Law. Individual bishops have a more limited authority to rule their diocese (they must always be loyal to, and in harmony with, the Magisterium–CCC 895), and national colleges of bishops are given a certain amount of power over various disciplinary and practical items, though never doctrinal.
Peter was given a special position of authority over the other Apostles and over the Church (Matt. 16:19; John 21:15-17). His successor, the Bishop of Rome, or Pope, continues Peter's ministry as Vicar of Christ, or visible head of Christ's Church (CCC 881–Christ is the invisible head since He is no longer on Earth). The Pope has "full, supreme, and universal power over the whole Church, which he can always exercise uninhindered" (CCC 882). The body of bishops has authority over the Church, but only when united with the Pope, and with his agreement (CCC 883-884). The Pope is aided in his mission by the Roman Curia, which consists of Cardinals and other officials who hold various offices and serve in various capacities, aiding the Pope in teaching and ruling the Church.
The Catholic Church is also infallible, meaning when the Church officially teaches a doctrine of faith or morals, it cannot err (CCC 890). This can be done by the body of bishops when they maintain the bond of unity among themselves, or when the Pope, as supreme shepherd of the faithful (ex cathedra) declares that a given doctrine regarding faith or morals must be held by all the faithful (CCC 891). Infallibility is also guaranteed in the dogmatic declarations of ecumenical councils, 21 of which have been held throughout the history of the Catholic Church. Some of the doctrines that are infallible, by these various means, include: purgatory, indulgences, the primacy and infallibility of the Pope, the Marian dogmas (Immaculate Conception, Assumption, perpetual virginity), the Trinity, the Redemption, invocation of saints and the veneration of them, their images, and their relics, original sin, the inerrancy and inspiration of Scripture, baptismal regeneration, the Real Presence of Christ in the Eucharist, the existence of angels and demons, the sacrificial nature of the Mass, and many more. The Church has never taught error, because it is the "pillar and foundation of truth" (1 Tim. 3:15).
St. Cyprian, a third century Church Father said: "outside of the Church, no salvation" (CCC 846). This is an infallible teaching of the Catholic Church. But rather than interpreting it literally to mean that one must be a card-carrying Catholic to be saved (CCC 819), the phrase refers to the fact that one must be connected to the Catholic Church in some way in order to be saved. Non-Catholics may be saved as well if they are invincibly ignorant of the fact that the Catholic Church is Christ's Church (CCC 847-848), but love God and serve Him faithfully; for by doing so, they can be members of the Church by desire (remember, mortal sin can be forgiven aside from the sacraments–CCC 1452).
All those who are baptized and profess the Catholic faith in its entirety are full-fledged members of the Catholic Church. However, there are many who have been separated from the Catholic Church by heresy, schism, and apostasy (CCC 2089). One is separated by heresy when he professes to be Christian, but rejects one or more infallible or definitive teachings of the Catholic Church, after baptism. One is separated by schism when he accepts all the teachings of the Catholic Church, but refuses to submit to the authority of the Pope. One is separated by apostasy when he rejects the Christian faith altogether. Those who are members of the Catholic Church in the full sense, however, can be members of the Church "in body" and not "in heart" (CCC 837), by sinning and not persevering in charity.