Short answers, pulled from the story.
Justice, in its broadest sense, is the treatment of individuals fairly, a society in which people receive what they deserve. The Institutes of Justinian, a 6th-century codification of Roman law, defines it as "the constant and perpetual will to render to each his due", which the Stanford Encyclopedia of Philosophy names as the most plausible core definition.
Plato defined justice as balance and harmony, the right relationship between conflicting aspects within an individual or a community. He held that a just person contributes to society according to their unique abilities and receives what is proportionate to their contribution, illustrated through the metaphor of a chariot whose charioteer, reason, controls the horses of spirit and desire.
Retributive justice holds that all guilty individuals deserve punishment proportional to the crime, and it is impartial with a scale of appropriateness. Restorative justice instead repairs the harm done to victims, encouraging dialogue and offender responsibility; it shows the highest rates of victim satisfaction and offender accountability but no improvement in recidivism.
In A Theory of Justice, John Rawls asks us to imagine choosing a theory of justice behind a veil of ignorance that hides our social status, wealth, talents, and life plans. Because we cannot bias the decision in our own favor, the choice models fairness, leading to two principles guaranteeing equal basic liberties and arranging inequalities to benefit the least advantaged.
The Euthyphro dilemma, posed by Plato in his dialogue Euthyphro, asks whether something is morally good because the gods command it, or whether the gods command it because it is already morally good. If the latter, justice lies beyond mortal understanding; if the former, morality exists independently of the gods and is subject to the judgment of mortals.
Legal theory names retribution, deterrence, rehabilitation, incapacitation, reparation, and denunciation as the main purposes of sentencing. These range from proportionate tariff sentences and long prison terms as examples to others, through individualized rehabilitative sentences and reparation schemes, to public punishment that expresses society's disapproval.