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.
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.