According to popular mythology, indulgences are a way to "buy your way out of purgatory." Indulgences frequently come up in attacks on the Catholic Church.
The medieval church did not actually sell indulgences, but that was an abuse that went on, unofficially. Similar to church fundraisers that you might attend today where there is a "suggested donation" price tag on things, the indulgence was for almsgiving, but functioned in practice as having been sold. Certainly, there was abuse there, and it was reformed by the Council of Trent.
Whereas the power of conferring Indulgences was granted by Christ to the Church; and she has, even in the most ancient times, used the said power, delivered unto her of God; the sacred holy Synod teaches, and enjoins, that the use of Indulgences, for the Christian people most salutary, and approved of by the authority of sacred Councils, is to be retained in the Church; and It condemns with anathema those who either assert, that they are useless; or who deny that there is in the Church the power of granting them. In granting them, however, It desires that, in accordance with the ancient and approved custom in the Church, moderation be observed; lest, by excessive facility, ecclesastical discipline be enervated. And being desirous that the abuses which have crept therein, and by occasion of which this honourable name of Indulgences is blasphemed by heretics, be amended and corrected, It ordains generally by this decree, that all evil gains for the obtaining thereof,--whence a most prolific cause of abuses amongst the Christian people has been derived,--be wholly abolished. But as regards the other abuses which have proceeded from superstition, ignorance, irreverence, or from what soever other source, since, by reason of the manifold corruptions in the places and provinces where the said abuses are committed, they cannot conveniently be specially prohibited; It commands all bishops, diligently to collect, each in his own church, all abuses of this nature, and to report them in the first provincial Synod; that, after having been reviewed by the opinions of the other bishops also, they may forthwith be referred to the Sovereign Roman Pontiff, by whose authority and prudence that which may be expedient for the universal Church will be ordained; that this the gift of holy Indulgences may be dispensed to all the faithful, piously, holily, and incorruptly.Indulgences were not abolished, however, and are still around in the church today. I would guess that the average Catholic thinks that they were done away with, like purgatory, after Vatican II. Obtaining an indulgence, like saying a Hail Mary, is not anything required of a Catholic, and you can be a good Catholic and never attempt to gain an indulgence.
I've found the best way to explain the doctrine behind indulgences is with a little story, so bear with me.
Suppose we have a family that is getting ready for a big Christmas feast. A little boy is dressed in his finest clothes, but there is still an hour before the guests arrive, and he begs his mother to go outside and play. His mother says he might, but makes him promise that he must NOT play in the mud! The boy promises, and out he goes. But, like my little 5 year old boy, he soon forgets, and when his mother comes to call him, she finds he is covered in mud. Now the boy is very sorry, and he begs forgiveness from his mother. And she immediately forgives him. But, because he is covered in mud, he needs to take a nice hot bath to be cleansed of the mud before he can join in the feast. The bath doesn't effect his forgiveness, because he was immediately forgiven. It just removes the results of his sin, so that he can enter into the feast purified.
So that is meant to explain purgatory, but it doesn't really tie indulgences in yet. Indulgences would be if the boy saw that he had done wrong, and tried to wash himself off with the garden hose, as a way of showing his mother how sorry he is. It doesn't effect his forgiveness, because he will be completely forgiven either way. But it might make for a shorter bath.
We call the bath, or purgatory, "temporal punishment." It isn't a punishment that is supposed to earn our salvation, because Jesus did all of that for us already. It is supposed to be part of a cleansing process that removes the stain of sin from our souls, because nothing impure can enter heaven.
The technical definition of an indulgences is "a remission of the temporal punishment due to sin, the guilt of which has been forgiven." This shows that the Catholic Church is not teaching that "Christ was not enough." We recognize that our guilt has already been forgiven through the death and resurrection of Christ.
Clear as mud? Let some better apologists than myself explain it.
A nice basic primer on indulgences
Catholic Answer's Myths About Indulgences
Dave Armstrong gives biblical evidence for indulgences
Feeling brave? Here's the official doctrine on indulgences from the Enchiridion.