Purgatory is being made ready. Not a second chance, and not a smaller hell. A person who dies in God's friendship, but still carrying the marks of what they did and what was done to them, is made whole before seeing God face to face. Nothing is decided there. What is decided has been decided; what remains is the healing.
Paul puts it in one image:
Every man's work shall be manifest; for the day of the Lord shall declare it, because it shall be revealed in fire; and the fire shall try every man's work, of what sort it is. If any man's work abide, which he hath built thereupon, he shall receive a reward. If any man's work burn, he shall suffer loss; but he himself shall be saved, yet so as by fire.
1 Corinthians 3:13–15, Douay-Rheims
Saved, yet so as by fire. The fire is not punishment for its own sake. It is what burns off the part that cannot come in.
The practice is older than the word. In the second book of Maccabees, after a battle, Judas Maccabeus takes up a collection and has sacrifice offered in Jerusalem for the soldiers who fell, so that they might be loosed from their sins. The book calls it a holy and wholesome thought to pray for the dead. Christians did it from the beginning: the inscriptions in the Roman catacombs ask for rest and refreshment for the person buried there, and Augustine ends the story of his mother's death by asking everyone who reads it to remember her at the altar. His page is here.
The Church's teaching is short and is in the Catechism, paragraphs 1030 to 1032: those being purified are helped by our prayer, and above all by the Mass.
The dead are not gone from the family. They are still in it, and they can still be prayed for, which means there is still something you can do for your mother, your husband, the friend who died too young. That is most of what purgatory means in practice: the door is not closed.
The oldest prayer for it is one sentence: Eternal rest grant unto them, O Lord, and let perpetual light shine upon them. May they rest in peace. Psalm 129, the De Profundis, has been said for the dead since the Church began; it is on the night page. The novena for the holy souls is nine days of it, and November is the month the whole Church gives to them.