Saints, broadly speaking, are those who follow Jesus Christ and live their lives according to his teaching. Catholics, however, also use the term narrowly to refer to especially holy men and women who, through extraordinary lives of virtue, have already entered Heaven.
Sainthood in the New Testament
The word saint literally means "holy," and, in the New Testament, saint referred to all who believed in Jesus Christ and followed his teachings. Saint Paul often addressed his epistles to "the saints" of a particular city (see, for instance, Ephesians 1:1 and 2 Corinthians 1:1), and the Acts of the Apostles talks about Saint Peter going to visit the saints in Lydda (Acts 9:32). The assumption was that those who followed Christ had been so transformed that they were now different from other men and women and, thus, should be considered holy.
Practitioners of Heroic Virtue
Very early on, however, the meaning of the word began to change. As Christianity began to spread, it became clear that some Christians lived lives of extraordinary, or heroic, virtue. While other Christians struggled to live out the gospel of Christ, these people were eminent examples of the moral virtues, and they easily practiced the theological virtues of faith, hope, and charity.
The word saint thus became more narrowly applied to such people, who were venerated after their deaths as saints, usually by the members of their local church or the Christians in the region where they had lived, because they were familiar with their good deeds. Eventually, the Catholic Church created a process, called "canonization," through which such venerable people could be recognized as saints by all Christians everywhere.
Canonized and Acclaimed Saints
Most of the saints whom we refer to by that title (for instance, St. Elizabeth Ann Seton) have gone through this process of canonization. Others, such as Saint Peter and Saint Paul, received the title through acclamation, or the universal recognition of their holiness.
Catholics believe that both types of saints (canonized and acclaimed) are already in Heaven, which is why one of the requirements for the canonization process is proof of miracles performed by the possible saint after his death. Canonized saints can be venerated anywhere and prayed to publicly, and their lives are held up to Christians still struggling here on earth as examples to be imitated.