Yes. Purgatory exists. The other three of the Four Last Things are Death, Heaven, and Hell.
The idea is a consoling thought. If "not one jot, not one tittle" will go unpunished, it is a huge help that we have an opportunity to repair before going to Heaven.
The Baltimore Catechism (BC) teaches the purgatory catechism answers.
What are the rewards or punishments appointed for men's souls after the Particular Judgment?
The Catholic catechism teaches that "the rewards or punishments appointed for men's souls after the Particular Judgment are Heaven, Purgatory, and Hell". (BC No. 1, Q. 412)
What is Hell?
Hell is a state to which the wicked are condemned, and in which they are deprived of the sight of God for all eternity, and are in dreadful torments. (BC No. 1, Q. 413)
What is Purgatory?
Purgatory is the state in which those suffer for a time who die guilty of venial sins, or without having satisfied for the punishment due to their sins. (BC No. 1, Q. 414)
What is Heaven?
Heaven is the state of everlasting life in which we see God face to face, are made like unto Him in glory, and enjoy eternal happiness. (BC No. 1, Q. 420)
Not only does the Catholic catechism teach it, but also the Bible.
The Douay-Rheims Bible says many times that "God will render to every man according to his works ... and shall not escape without punishment."
See these Bible quotes: Matt. 5:25, 26; Matt. 12:32, 36; Apoc. 21:27; Luke 12:58, 59; 1Cor. 3:13-15; 1Pet. 3: 18-20.
What is Original Sin?
Original Sin is the sin of our first parents, Adam and Eve.
What is sin?
Sin is disobedience to God's laws.
After the Creation (Gen. 1:1, Douay-Rheims Bible), Adam and Eve chose to do the very one thing God had forbidden. The effects are upon all their children. Where is their hope?
God promised to send His only Son, Jesus (Gen 3:15, Douay-Rheims Bible), through the Blessed Virgin Mary, Mediatrix of All Graces! Jesus established His Church on earth and gave us the Sacrament of Baptism by which we are cleansed of Original Sin and by which we profess the Catholic religion.
What is actual sin?
Actual sin is any sin we ourselves commit.
Uh-oh. What do we do for the sins we have committed after Baptism? How can we get rid of the guilt due to actual sin? We're not just saved by Baptism because we can still offend God. So what's to do?
Enter: the Sacrament of Penance. By confessing our actual sins to a priest to gain absolution we can have the grace to avoid Hell for any mortal sins and the grace to avoid punishment in purgatory for venial sins. The more sorry we are that we have offended God, the more grace we receive.
If we cannot get to confession right away, we can make an act of contrition or an act of sorrow. Here is a beautiful version of the perfect Act of Contrition Prayer.
For mortal sins we also need to make the intention to go to sacramental confession as soon as possible. Yes, this intention "covers" us in the meantime in case of accidental or unprovided death, although it may not be enough to remove the need to go to purgatory. This is why it is important to have the prompt habit of making an act of contrition right away.
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