What is an Indulgence

The Decree of Indulgence for Divine Mercy Sunday grants a plenary or full indulgence to those who satisfy certain conditions established by the Church and a partial (incomplete) indulgence to those who fulfill some but not all or the conditions.

A plenary indulgence means that by the merits of Jesus Christ, the Blessed Virgin Mary and all the saints, the full remission of the temporal punishment due to sacramentally forgiven sins is obtained. The person becomes as if just baptized and would fly immediately to heaven if he died in that instant. A partial indulgence means that a portion of the temporal punishment due to forgiven sin is remitted. Partial indulgences are received either by doing some act to which a partial indulgence is attached (e.g. praying a partially indulgenced prayer), or by the incomplete fulfillment of the conditions attached to a plenary indulgence. 

Eternal and Temporal Punishment or Guilt
There are two kinds of punishment attached to sin, eternal and temporal. If the sin is mortal (serious, grave) sin, the person loses the friendship of God and with it the life of divine grace within. This punishment is eternal. If the person is not restored to grace before death he will be punished forever in hell, since serious sin is an infinite insult to an All-Holy God and thus deserves a like punishment. It was to repair for such sin that Jesus became man and was crucified. As God His sacrifice was infinitely meritorious, as Man He was able to represent us. He thus could expiate for our mortal sins, which are not just beyond our power of expiation but infinitely beyond it.
Mortal sin, and also venial sin (which has no eternal punishment attached to it), both disturb the right order within us and in the order of justice in general. We all experience these temporal (or in-time, in-this-world) consequences of sin, both both personally and socially. Sin changes us (or rather we sin because we are not what we are supposed to be), and like a pebble in a pond these changes have  effects beyond us. Not only must we be sorry for our sins, but we must be more thoroughly converted to the Lord, and demonstrate that conversion (Acts 26:20) by our actions. So, while sacramental absolution forgives the eternal guilt of sin, which requires the infinite merits of Christ, it does not necessarily remove all the temporal punishment, since they are somewhat within our power to repair (and somewhat unknown to us). Depending on our degree of sorrow, absolution may result in the expiation of all the temporal guilt of sin. However, for that which it does not repair, we must offer further expiation through prayer, penance, carrying the Cross etc., or after death be purified in purgatory (Rev 21:27). 

What an Indulgence does is to take an occasion of such expiation (a certain prayer, penance, charity or other designated work) and add to its intrinsic merit  before God an additional value based on the treasury of merits of Jesus Christ, and those perfectly united to Him in heaven (the saints). This can either partially, or under certain conditions, totally remit the temporal punishment due to sin. This depends, naturally, on our openness to God's grace. A mechanical performance of an indulgenced work would not have effect. Performing an indulgenced work should have the consequence of fixing our will away from our sins and entirely on God. This is why among the most important of the conditions for receiving a plenary indulgence, and the hardest to satisfy, is the complete detachment or detestation of our sins. By detesting our sins we orient our will away from creatures (to the degree we love them inordinately), towards God. In this way we open our will to the action of His mercy flowing into our souls, which alone is able to effect the complete remission of the temporal punishment to our sins. 
An example will perhaps better illustrate these points. A boy playing ball breaks a window of his home. Contrite and sorrowful he goes to his father, who forgives him. However, despite the forgiveness the window is still broken and must be repaired. Since the boy's personal resources are insufficient to pay for a new window, the father requires him to pay a few dollars from his savings and forego some of his allowance for several weeks, but that he, the father, will pay the rest. This balances justice and mercy (generous love). To ask the boy to do nothing, when it is possible for him to make some reparation, would not be in accordance  with the truth, or even the boy's good. Yet, even this temporal debt is beyond the boy's possibilities. Therefore, from his own treasury the father generously makes up what the child cannot provide. This is indulgence. Unlike the theologies that say "we are washed it the blood of the Lamb and there is nothing left to do," Catholic teaching respects the natural order of justice, as Jesus clearly did in the Gospels, yet recognizes that man cannot foresee or undo all the temporal  consequences of his sin. However, God in His mercy will satisfy justice for what we cannot repair.

Note on Partial Indulgences (days and years)
In the past partial indulgences were "counted" in days (e.g. 300 days) or years (e.g. 5 years). Catholics often mistakenly thought that this meant "time off of purgatory." Since there is no time in purgatory, as we understand it, it meant instead the remission of temporal punishment analogous to a certain amount of penitence as practiced in the early Church. This was a very generous standard, since the penitence required for sacramental absolution in the early centuries was arduous, indeed. However, with Pope Paul VI's 1968 revision of the Enchiridion Indulgentiarum (Collection or Handbook of Indulgences), this confusing way of counting partial indulgences was suppressed, and the evaluation of a partial indulgence left to God. 
There are many prayers still circulating on prayer cards and in prayer books which have partial indulgences in days and years attached to them. However, all grants of indulgence issued prior to 1968, unless re-issued in the Enchiridion or specifically exempted by papal decree or privilege, were suppressed by Pope Paul VI. Thus, these many specific prayers with their attached indulgences, as well as the manner of measuring partial indulgences, are no longer valid. Some of them may still receive an indulgence, though, because of being re-issued in the new Enchiridion (e.g. the Anima Christi, the Prayer before a Crucifix and many other formal prayers). All other prayers previously indulgenced could, nonetheless, receive a partial indulgence under the general grants of indulgence which Pope Paul VI, and Pope John Paul II in his 1999 revision of the Enchiridion, established. These general grants establish partial indulgences for devout prayer, penitence and charity, and are a new and very generous inclusion in the Church's grants of indulgence. They have made it unnecessary to grant specific indulgences to prayers and other pious acts, as was done in the past.