Short answers, pulled from the story.
Purgatory in Catholic theology is a transitional state after death in which souls who died in God's grace but are still imperfectly purified are cleansed before entering heaven. Catholic doctrine, first formally defined at the Second Council of Lyon in 1274, holds that such souls benefit from the prayers, Masses, and alms of the living. The Church teaches that purgatory indicates a condition of existence, not a physical place.
The Catholic Church formally defined its teaching on purgatory for the first time at the Second Council of Lyon in 1274. The Council of Florence (1431-1449) and the Council of Trent, in its decree of the 4th of December 1563, each repeated the same two core points: that some saved souls require purification after death, and that the prayers and pious acts of the living assist those souls.
Catherine of Genoa (1447-1510) reframed purgatory as a voluntary, loving, and joyful inner fire rather than an external punishment in a physical place. She described the soul still bound to desires derived from sin as unable to enjoy the beatific vision of God, with love for God itself becoming the flame that purifies it. Pope Benedict XVI recommended her presentation to theologians in his 2007 encyclical Spe salvi.
Eastern Orthodox Christians acknowledge an intermediate state after death and hold that prayers for the dead are efficacious, but they reject the Catholic term purgatory and its associated imagery of fire and a fixed place. At the Council of Florence (1431-1449), Orthodox representatives stated that cleansing after death occurs, but not by some purifying fire and particular punishments in some place. The state in which souls undergo this experience is generally referred to as Hades.
The word purgatory comes from the Latin purgatorium, meaning a place of cleansing, from the verb purgo, to clean. It entered English via Anglo-Norman and Old French. The noun purgatorium appeared perhaps only between 1160 and 1180, giving rise to speaking of purgatory as a place rather than merely a process; before that, only the adjective form purgatorius, as in purgatorius ignis (cleansing fire), existed.
Gehenna in Judaism functions similarly to purgatory as a place of after-death purification for most sinners. The school of R. Akiba held that the period lasts twelve months, while R. Johanan b. Nuri argued for forty-nine days, each based on different readings of Isaiah 66:23-24. Maimonides, however, declared that the descriptions of Gehenna as a place of punishment were pedagogically motivated inventions to encourage respect of Torah commandments, and that the souls of the wicked would actually be annihilated rather than sent to Gehenna.