For the Catholic Purgatory is a period of purification after death.
When we die, our souls are judged immediately by Christ in what's called the "Particular Judgment":
Each man receives his eternal retribution in his immortal soul at the very moment of his death, in a particular judgment that refers his life to Christ: either entrance into the blessedness of heaven — through a purification or immediately, — or immediate and everlasting damnation. (_Catechism_, 1022)
Purgatory is this period of purification before heaven.
It's not always well understood by today's Catholics but Purgatory is still very much a part of Catholic doctrine.
It is not a "second chance"
Don't think that Purgatory is anything like a "second chance" for those who have not won the reward of heaven!
During our human life, we either accept or reject God's offer of divine grace. Once we die, our choice is definitive. We cannot change our mind after death. (Catechism, 1021)
Heaven and hell are real. They're part of a viewpoint that's fully Catholic and Purgatory is simply a transitional state for those who have merited heaven but still have aspects of their souls that are not yet fully purified. Purgatory is where that purification happens after death.
The souls in Purgatory are assured of salvation. They've died in God's grace and friendship, and will end up in heaven. But they're not yet in a full state of holiness — the holiness that's necessary to behold God "face to face" in heaven. (Catechism, 1030)
Basis in Scripture and Tradition
The Catholic Church is often accused of inventing the concept of Purgatory out of thin air. Not so!
You don't hear about it from many who aren't Catholic but Purgatory does have deep roots in Sacred Scripture as well as Catholic Tradition — the full, living faith of the Apostles as received from Christ.
First, it's based on the ancient Jewish practice of prayer for the dead, as mentioned in Scripture: "Therefore [Judas Maccabeus] made atonement for the dead, that they might be delivered from their sin." (2 Macc 12:46)
The early Christians continued this practice: "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." (Catechism, 1032)
Inscriptions on the walls and tombs of the Catacombs testify to the belief of many early Catholics in Purgatory.
The words of the Apostles in the New Testament also clearly tell us about being "tested by fire" (1 Pet 1:7). St. Paul warns us that if someone builds on the true foundation of Christ but doesn't take care to build well, "the person will be saved, but only as through fire" (1 Cor 3:15).
Finally, the Catechism quotes St. Gregory the Great:
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. (_Catechism_, 1031)
(It's worth reading the Catechism's brief section on Catholic Purgatory to see the straightforward teaching of Catholics about Purgatory.)
Purgatory: part of the Good News
Part of the faith of Catholics is that Purgatory is a good thing!
Purgatory reveals the depth of God's mercy: even those who are not yet perfect can attain the fullness of heaven.
For Catholics Purgatory helps us hope in perfection even when we can't completely achieve it in this life.