Indulgences are part of the Catholic doctrine of Purgatory.
Specifically, this doctrine states that a person must have the last vestiges of venial sin purged from their soul before entering heaven. Purgatory, then, is a state of the purgation, which removes the last impurities from the soul. It is the last part of Christ's remission of an individual's sins, the first two being Baptism and Confession.
This remaining effect of venial sin is sort of like somebody coming in from a motocross; they are covered with mud, and although they can strip off all their clothes, they're still not going to walk into the living room with new white carpeting until they shower first and get the mud and dirty water off their face, hands, and hair. Same thing with Purgatory. You are a sinner. Redeemed, yes, but still a sinner. When you die, your sins have all been remitted by Christ, it's true---but you still remain a sinner, and so Christ has provided you with this purgational process to cleanse you of all effects and inclination to sin before you eneter heaven, so you enter it perfectly pure.
An indulgence is a full or partial remission of a part of this purgation process, applied to an individual who is in Purgatory. The doctrine grew out of the Jewish practice of saying certain prayers every day for a year after the death of a lived one, with the belief that such prayers helped to purify the soul of the loved one and hastened the loved one's soul to heaven; you can read more about this if you type "Kaddish" into your browser. I'll see if I can find an old link later.
So basically, if you say some prayers or perform a good work, with the intention that the spiritual benefit of those prayers or works are to be applied to the poor souls in Purgatory, then God, Who counts these works as part of the "treasury of graces", will duly apply them to those souls, thus shortening their stay in Purgatory.
The Church believes that one person can perform spiritual action that can be applied to others, and which are beneficial to others. This is exactly what Christ did for us---He died on the Cross, not for himself, but to apply His actions to us.
In the Middle Ages, indulgences were abused; people were told if they paid a certain amount of money, they could gain an indulgence. This was a violation of the teaching, and after the Council of Trent, it was made clear that indulgences could not be bought or sold. The sale of indulgence was what (rightly) set Martin Luther off. Since Trent, however, indulgences have been obtained entirely through spiritual means, and anyone who tells you that the Church uses them to obtain money is about 500 years out of date. Protestant televangelists who see giant Jesuses bilk more money out of people than the Catholic Church does with indulgences (or anyhting else, I might add).
You don't hear a lot about indulgences any more, but if nothing else, at the very least they encourage people to perform good works, instead of sins, and that in itself is a good thing.
