The following is from the official Catechism of the Catholic Church. Please read carefully. The paragraphs below dispel many Protestant (and Eastern Orthodox) misconceptions about Purgatory:
1030 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.So first of all, Purgatory is not eternal Hell. Secondly, only the elect, that is saved Christians, will go there. Purgatory is a place only for those on their way to Heaven. It is the final purification of those who die in fellowship with Christ.
1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned. The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come. (St. Gregory the Great, Dial. 4,39:PL 77,396; cf. Mt 12:31.)
This teaching is also based on the practice of prayer for the dead, already mentioned in Sacred Scripture: "Therefore [Judas Maccabeus] made atonement for the dead, that they might be delivered from their sin." From the beginning the Church has honored the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God. The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead:
Let us help and commemorate them. If Job's sons were purified by their father's sacrifice, 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. (611 St. John Chrysostom, Hom. in 1 Cor. 41,5:PG 61,361; cf. Job 1:5.)
There are Scripture passages relating to prayer for the dead. If one accepts 2 Maccabees (as quoted above) as canonical, then once must accept prayer for the dead. Many scholars believe that St Paul prays for a dead friend in 2 Timothy chapter 1:
 May the Lord grant mercy to the household of Onesiphorus, for he often refreshed me; he was not ashamed of my chains,
 but when he arrived in Rome he searched for me eagerly and found me --
 may the Lord grant him to find mercy from the Lord on that Day -- and you well know all the service he rendered at Ephesus.
Our Savior Christ also mentions that there is opportunity for forgiveness in this life and after death:
Matthew 12:32 And whoever says a word against the Son of man will be forgiven; but whoever speaks against the Holy Spirit will not be forgiven, either in this age or in the age to come.
But the most convincing passage for me was 1 Corinthians 3:13-15:
 each man's work will become manifest; for the Day will disclose it, because it will be revealed with fire, and the fire will test what sort of work each one has done.
 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.
First of all, each man will be judged and his work "will be revealed with fire." The good we have done will survive the fire and will be our "reward." The evil we have done will be "burned up" and "he will suffer loss, though he himself will be saved, but only as through fire."
Here we see that this kind of fire is not Hell, but "he himself will be saved, but only as through fire." The Greek word for fire is "pur" and it is the same Indo-European root from which "PUR-gatory" derives. Purgatory is that state of purification by fire for those who are already saved.
The Protestant might ask at this point, "Well if somebody is already saved, then why do that have to pass through this fire? Didn't Christ die for all their sins?"
Yes, Christ died for their sins and has redeemed them. But He died that we might become actually holy. "Be holy as I am holy." The fire of Purgatory is the fire of God's love causing us to "suffer loss" by a sort of final repentance from our sins. It is therefore painful because we must let go of the desires of the flesh and face our failings. This is what it means to "suffer loss". We can't get around the words of St Paul who says that Christians must pass through fire after death.
If Uzzah was killed by God for merely touching the Ark of the Covenant, then we must be fully sanctified to enter Heaven. The debt has been paid but we have not fully been transformed into the image of Christ. Christ died to make us actually and really holy. Purgatory is this final transformation by which our Christ-centered actions are acknowledged and our sinful affections are burned away.
1 Cor 12:26 If one member suffers, all suffer together; if one member is honored, all rejoice together.
Best of all, we should have the Holy Sacrifice of the Mass offered for the faithful departed. We are a family and we're all in this together. One thing I'm learning as a Catholic is that we really are one huge family in Christ. When we pray for the departed we are asking God "who is a consuming fire" to assist those who are undergoing their final repentance and purification as they prepare to entere His All-Holy presence.
I commend to you two of my recent blog posts:
Orthodox Judaism and Purgatory
CS Lewis and Purgatory