Purgatory is a teaching of the Church which is too often forgotten. When people sin, it requires reparation to God. Even though He forgives them and Jesus' sacrifice on the cross made up for the debt to God created by the sin, there is still damage to the soul that must be repaired. Every sin causes a little bit of damage to the soul and makes it less pure. In order to enter Heaven, a person's soul must be completely pure, because God is perfect and cannot have anything tainted in His presence. Purgatory is that place where souls are purified so that they can go to Heaven. The souls in Purgatory suffer because they are being purified of any small imperfections which remain on their soul, but they also are joyful, because they know that they will reach Heaven sometime in the future. Their entrance into Heaven is guaranteed. We can offer prayers for the souls in Purgatory that their time in that place be shortened. Our prayers to God on their behalf help to speed up the time of purification so that they can get to Heaven even sooner.
In the Bible – Purgatory Exists and Pray for them
There is no place in the Bible in which Purgatory is specifically mentioned by name, but the idea of it is supported in the Bible as well as the practice of praying for the souls of those who have died. “If the work which any man has built on the foundation survives, he will receive a reward. If any man's work is burned up, he will suffer loss, though he himself will be saved, but only as through fire.” - 1 Corinthians 3: 14 – 15. St. Paul speaks of the life of a person, and whether they will be found worthy of entering heaven after their life is over. He mentions that even if a person's life is not perfect, he can still be saved, but must be purified with fire. In the Old Testament, there is mention of praying for the dead. “For if he were not expecting that those who had fallen would rise again, it would have been superfluous and foolish to pray for the dead. But if he was looking to the splendid reward that is laid up for those who fall asleep in godliness, it was a holy and pious thought. Therefore he made atonement for the dead, that they might be delivered from their sin.” - 2 Maccabees 12: 44 – 45. This refers to Judas Maccabeus who found some dead soldiers who had committed sins of idolatry. He had sacrifices offered for them for the atonement of their sin, after their death. As the passage points out, he clearly thought they could be saved, but he also knew that they needed to be purified of their sin. The bible records this as a holy and pious thought.
Early Church Fathers – Pray for the Dead
The early Church also understood about Purgatory and praying for the dead, particularly at Mass. They knew that this was an act of mercy and charity.
We offer sacrifices for the dead on their birthday anniversaries [the date of death—birth into eternal life] – Tertullian The Crown 3:3
A woman, after the death of her husband...prays for his soul and asks that he may, while waiting, find rest; and that he may share in the first resurrection. And each year, on the anniversary of his death, she offers the sacrifice – Tertullian Monogamy10:1–2
Then we make mention also of those who have already fallen asleep: first, the patriarchs, prophets, apostles, and martyrs, that through their prayers and supplications God would receive our petition; next, we make mention also of the holy fathers and bishops who have already fallen asleep, and, to put it simply, of all among us who have already fallen asleep, for we believe that it will be of very great benefit to the souls of those for whom the petition is carried up, while this holy and most solemn sacrifice is laid out – Cyril of Jerusalem Catechetical Lectures 23:5:9
Let us help and commemorate them. If Job’s sons were purified by their father’s sacrifice (Job 1:5), why would we doubt that our offerings for the dead bring them some consolation? Let us not hesitate to help those who have died and to offer our prayers for them. - John Chrysostom Homilies on First Corinthians 41.5
There is an ecclesiastical discipline, as the faithful know, when the names of the martyrs are read aloud in that place at the altar of God, where prayer is not offered for them. Prayer, however, is offered for other dead who are remembered. It is wrong to pray for a martyr, to whose prayers we ought ourselves be commended. - Augustine Sermons 159:1
Temporal punishments are suffered by some in this life only, by some after death, by some both here and hereafter, but all of them before that last and strictest judgment. But not all who suffer temporal punishments after death will come to eternal punishments, which are to follow after that judgment. - Augustine The City of God 21:13
If a man departs this life with lighter faults, he is condemned to fire which burns away the lighter materials, and prepares the soul for the kingdom of God, where nothing defiled may enter. For if on the foundation of Christ you have built not only gold and silver and precious stones (I Cor., 3); but also wood and hay and stubble, what do you expect when the soul shall be separated from the body? Would you enter into heaven with your wood and hay and stubble and thus defile the kingdom of God; or on account of these hindrances would you remain without and receive no reward for your gold and silver and precious stones? Neither is this just. It remains then that you be committed to the fire which will burn the light materials; for our God to those who can comprehend heavenly things is called a cleansing fire. But this fire consumes not the creature, but what the creature has himself built, wood, and hay and stubble. It is manifest that the fire destroys the wood of our transgressions and then returns to us the reward of our great works. - Origen Patres Groeci. XIII, col. 445, 448
These authors speak of the existence of a place after death which is not Heaven but which is for those who will eventually reach Heaven. It is particularly good to remember them and pray for them at mass. This practice is from the earliest days of the church, as evidenced by these passages.