An indulgence is the remission of punishment for a forgiven sin. In the Catholic Church the sacrament of reconciliation brings forgiveness of sins, which may still carry punishments.
Indulgences relax the punishments.
Indulgences first appeared in the 11th century as a way of reducing the penance one was required to perform after confessing a sin. Several hundred years later they were understood to relax punishments after death partially or totally. In the past, some indulgences came with a certain number of days or years attached. The church explained that if you gained the indulgence it would speed your progress from purgatory to heaven by that amount of time. Today we no longer apply this metaphor of time.