Martyr
The term martyr comes from the Greek word martuV which means "witness". Martyrdom is the  supreme witness given to the truth of the faith: it means bearing witness even unto death. The martyr bears witness to Christ who died and rose, to whom he is united by charity. He bears witness to the truth of the faith and of Christian doctrine. He endures death through an act of fortitude.

The faithful of the primitive Church believed that martyrs were perfect Christians and saints since they had shown the supreme proof of love by giving their lives for Christ; by their sufferings, they had attained eternal life and were indefectibly united to Christ, the Head of the Mystical Body. These reasons induced the Christians, still oppressed by persecution, to invoke the intercession of the martyrs. They begged them to intercede before God to obtain for the faithful on earth the grace to imitate the martyrs in the unquestioning and complete profession of faith.

Beatification
Beatification is the act by which the Church, through a papal decree, permits a specified diocese, region, nation, or religious institute, to honor with public cult under the title “Blessed” a Christian person who has died with a reputation for holiness. The cult usually consists of a Mass and Divine Office (liturgical prayers) in the person’s honor.

Formal beatification is a positive declaration, following a canonical process, that a person did practice heroic Christian virtue, or suffered a true martyrdom, and after death worked authentic miracles upon being invoked in prayer. Besides witnesses’ testimony to his virtues, evidence of two first-class miracles is required, though this requirement may be waived in the case of a martyr, the martyrdom being itself the miracle.

Equivalent beatification is the silent consent of the Church, aware of, yet not opposing, the public cult given to one of its children over a long period of time.

In proclaiming a person “Blessed”, the Pope does not exercise his infallibility but his magisterial authority, for he does not declare definitively that the person is in glory. Beatification, then, does not demand faith yet gives moral certainty of its truth, and to deny it would be temerarious.

Canonization
Canonization is the solemn act by which the Pope, with definitive sentence, inscribes in the catalogue of saints a Christian person who has previously been beatified. By this act he declares that the person placed on the altar now reigns in eternal glory and decrees that the Universal Church show him the honor due to a saint. The solemn canonization of a saint is an infallible and irrevocable decision of the Pope.

The purpose of Beatification and Canonization
The principal scope of beatification and canonization is to propose as models of Christian life those who knew how to live the Gospel in definite circumstances and environments. The pastoral reason why the cult of the “Blessed” is limited only to certain sectors of the Church is the existence of those persons whose lives and virtues may be good for a certain sector of the Church but inopportune to the others. For instance, the good examples of founders of Religious communities may only be opportune for the members of their respective communities and not necessarily for others. There are also some persons whose fame of holiness is only known in a certain region and so his or her cult need not be “imposed” on other regions. When one is canonized, it can mean that his or her fame of sanctity is universal and that his or her exemplary life is opportune to all the members of the Church.