“All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven.”
Catechism of the Catholic Church; Paragraph 1030
“The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.”
Catechism of the Catholic Church; Paragraph 1031
Where is it in the Bible?
Dave Armstrong has a very thorough article on this topic. We, however, will only look at a few convincing examples from Scripture:
- Matthew 12:32: “And whosoever shall speak a word against the Son of man, it shall be forgiven him: but he that shall speak against the Holy Ghost, it shall not be forgiven him neither in this world, nor in the world to come” (DR Version)
- This argument, that Jesus here implicitly refers to some purification of sin that occurs "in the world to come," is used by Saint Isidore of Seville, Saint Augustine, Saint Gregory the Great, Venerable Bede, and Saint Bernard. As Father John Hardon explains, “by this it is to be understood that certain faults are pardoned in this life, and certain others in the life to come.” This is implicit in Jesus’s statement and seems to match perfectly with the Catholic Church's doctrine on Purgatory.
- Here, we see Saint Paul speaking at length about our works (building upon the foundation) and how they will be treated during Judgment. He claims that our works will be “revealed with fire, and the fire itself will test the quality of each one’s work.” The commentary in the New American Bible explains that fire “both destroys and purifies.” It is the final verse, verse 15, that implicitly points to Purgatory. “If someone’s work is burned up,” if it is purified by the fire so that “the person will be saved,” then the fire is purgative so that the person may be purified for heaven. See the Catechism paragraph 1030 (cited above) and how it meshes perfectly with what Saint Paul is saying to the Corinthians.
- The doctrine of Purgatory is closely related to the practice of prayers for the dead; if there was no Purgatory, there would be no reason to pray for the deceased. Though 2 Maccabees is a deutero-canonical book, and only in the Catholic canon, there is still much to be said about it in convincing Protestants. Clearly, as the author points out, this ritual of “praying for the dead” is a “very excellent and noble” practice of the ancient Jews. If this practice were appalling to Jesus, it is likely that he would have corrected the Jews,  as he did with many of the other Jewish traditions. Jesus does not correct this Jewish tradition, and if this were one of the things Jesus said but was not written down (John 21:25), it is likely a prohibition of prayers for the dead would have appeared in the writings of the early Church. Instead, the Church Fathers consistently urge people to pray for those in Purgatory.
- 1 John 1:8—When we die, we are imperfect: “If we say, ‘We are without sin,’ we deceive ourselves,and the truth is not in us”
- Revelation 21:27—“In Heaven, we will all be perfect:” “I saw no temple in the city, for its temple is the Lord God almighty and the Lamb. The city had no need of sun or moon to shine on it, for the glory of God gave it light, and its lamp was the Lamb. The nations will walk by its light, and to it the kings of the earth will bring their treasure. During the day its gates will never be shut, and there will be no night there. The treasure and wealth of the nations will be brought there, but nothing unclean will enter it, nor any[one] who does abominable things or tells lies. Only those will enter whose names are written in the Lamb’s book of life.”
- “Put these two principles together, and purgatory necessarily follows.”
The Church does not profess faith in Purgatory apart from evidence in the Scriptures. In fact, she often uses them to explain and defend her faith in this purifying state of existence.
 Edward Hanna; Catholic Encyclopedia; Purgatory
 John Hardon; The Catholic Catechism
 New American Bible; Note on 1 Corinthians 3:13
 Dave Armstrong’s Biblical Evidence for Catholicism Blog; Biblical Evidence for Purgatory: 25 Bible Passages
 Peter Kreeft; Catholic Christianity