What is an Indulgence?
An indulgence is defined as "the remission before God of the temporal punishment due for sins already forgiven as far as their guilt is concerned." The first thing to note is that forgiveness of a sin is separate from punishment for the sin. Through sacramental confession we obtain forgiveness, but we aren't let off the hook as far as punishment goes.
There are two types of Indulgence: Partial and Plenary.
Partial Indulgence
A partial indulgence removes part of the temporal punishment due to sins.
Plenary Indulgence
A plenary indulgence removes all of it. This punishment may come either in this life, in the form of various sufferings, or in the next life, in purgatory. What we don't get rid of here we suffer there.
(Source: Catholic.org)