Why do Catholic believe in Purgatory? Is this mentioned in the Bible?

The main body of Christians have always believed in the existence of a place between Heaven and Hell where souls go to be punished for lesser sins and to repay the debt of temporal punishment for sins which have been forgiven. Even after Moses was forgiven by God, he was still punished for his sin. (2 Kg. or 2 Sam. 12:13-14). The primitive Church Fathers regarded the doctrine of Purgatory as one of the basic tenets of the Christian faith. St. Augustine, one of the greatest doctors of the Church, said the doctrine of Purgatory "has been received from the Fathers and it is observed by the Universal Church." True, the word "Purgatory" does not appear in the Bible, but a place where lesser sins are purged away and the soul is saved "yet so as by fire," is mentioned (1 Cor. 3:15). Also, the Bible distinguishes between those who enter Heaven straightaway, calling them "the church of the firstborn" (Heb. 12:23), and those who enter after having undergone a purgation, calling them "the spirits of the just made perfect." (Heb. 12:23). Christ Himself stated: "Amen I say to thee, thou shalt not go out from thence till thou repay the last farthing." (Matt. 5 :26). And: "Every idle word that men shall speak, they shall render an account for it in the day of judgment." (Matt. 12:36). These are obviously references to Purgatory. Further, the Second Book of Machabees (which was dropped from the Scriptures by the Protestant Reformers) says: "It is therefore a holy and wholesome thought to pray for the dead, that they may be loosed from sins." (2 Mach. 12:46). Ancient Christian tomb inscriptions from the second and third centuries frequently contain an appeal for prayers for the dead. In fact, the custom of praying for the dead – which is meaningless if there is no Purgatory – was universal among Christians for the fifteen centuries preceding the Protestant Reformation.

Furthermore, ordinary justice calls for a place of purgation between Heaven and Hell. Take our own courts of justice, for example. For major crimes a person is executed or sentenced to life imprisonment (Hell); for minor crimes a person is sentenced to temporary imprisonment for punishment and rehabilitation (Purgatory); for no crime at all a person is rewarded with the blessing of free citizenship (Heaven). If a thief steals some money, then regrets his deed and asks the victim for forgiveness, it is quite just for the victim to forgive him yet still insist on restitution. God, who is infinitely just, insists on holy restitution. This is made either in this life, by doing penance (Matt. 3:2; Luke 3:8, 13:3; Apoc. 3:2-3, 19), or in Purgatory.

Also, what Christian is there who, despite his faith in Christ and his sincere attempts to be Christlike, does not find sin and worldliness still in his heart? "For in many things we all offend." (James 3:2). Yet "there shall not enter into it [the new Jerusalem, Heaven] anything defiled." (Apoc. or Rev. 21:27). In Purgatory the soul is mercifully purified of all stain; there God carries out the work of spiritual purification which most Christians neglected and resisted on earth. It is important to remember that Catholics do not believe that Christ simply covers over their sinful souls, like covering a manure heap with a blanket of snow (Martin Luther's description of God's forgiveness). Rather, Christ insists that we be truly holy and sinless to the core of our souls. "Be you therefore perfect, as also your heavenly Father is perfect." (Matt. 5:48). This growth in sinlessness – in Christian virtue and holiness – is of course the work of an entire lifetime (and is possible only through the grace of God). With many this cleansing is completed only in Purgatory. If there is no Purgatory, but only Heaven for the perfect and Hell for the imperfect, then the vast majority of us are hoping in vain for life eternal in Heaven.

http://www.olrl.org/apologetics/cathansr.shtml#ans17