We all do, at least implicitly. And I would wager that if more people understood what the Catholic Church means when it speaks of purgatory, they would find themselves compelled to also acknowledge its existence.
Too many people speak of purgatory as a "place," as they do heaven and hell. Here is how the Catechism of the Catholic Church defines it:
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.The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned ... The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire (1 Cor.3:15, 1 Peter 1:7) ...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" (2 Maccabees 12:46). 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 ... (CCC 1030-32)
Purgatory is a process, a process of cleansing. I go into this in more depth in The God Who is Love, but let me give a few points:
Jesus taught, “from within, out of the heart of man, come evil thoughts, fornication, theft, murder, adultery, coveting, wickedness, deceit, licentiousness, envy, slander, pride, foolishness. All these evil things come from within, and they defile a man” (Mk.7:21-23). Now hopefully we Christians are not committing murder but the unkind thoughts or harsh judgments we harbor toward others (the roots of murder) may still be very much with us. These are not acts so great as to completely severe our union with Christ, but they do inhibit the flow of His life within us. When we shed our bodies at death do the impurities, the imperfections in our capacity to love, just disappear? If they are in the heart, and we take that with us, then no.
Every Christian I know believes that we will be "perfect" in heaven. No more sinful actions and no more sinful movements of the heart. And we're right to believe so, Scripture teaches that “nothing unclean shall enter” the Heavenly Jerusalem (Rev.21:27). So if the vast majority of us have imperfect hearts/souls at the moment of death, and yet we will be perfect when fully united to God in heaven, a cleansing must take place! It's implicit in the understanding of all Christians; the Catholic Church is simply drawing our attention to it.
There's a lot more to be said. If you're curious you might want to check out Chapter 6 of The God Who is Love, "Confession and the Ongoing Nature of Salvation." Let me give the final word in this post to my Anglican brother, C.S. Lewis:
Our souls demand Purgatory, don’t they? Would it not break [our] heart if God said to us, “It is true, my son, that your breath smells and your rags drip with mud and slime, but we are charitable here and no one will upbraid you with these things, nor draw away from you. Enter into the joy”? Should we not reply, “With submission, sir, and if there is no objection, I’d rather be cleaned first.” “It may hurt, you know” - “Even so, sir.”