Retributive Justice:
For every society there are laws to follow. To keep society running smoothly you have to stop working with people that do not obey those laws.
This works well if there are only a few dissenters from society. But if too many people disobey, then you will stop working with everyone, and there will be no society anymore. Retributive justice creates an incentive to obey societal rules by punishing people who commit crimes.
Thus there are two parts to retributive justice, to stop working with people who won't obey the rules, and to give them an incentive to obey.
Contract Termination:
The first idea, to stop working with people, comes from the relationship ethics of the theory of neurological intelligence. The relationship ethics show that society's laws and ethics come from mutually beneficial social contracts. Therefore when someone breaks a contract you terminate it. Or if someone breaks a law you can stop giving them the benefits of the law. According to the theory of social contracts people benefit greatly from being a part of society, so cutting them off is punishment enough for their crimes.
This can apply to the rest of the relationship ethics as follows:
When someone refuses to contribute to society, then society does not have to contribute to them.
When someone refuses to negotiate fair laws they should be left out of negotiation.
When someone does not fulfill their responsibility then they should not be trusted with any responsibilities.
When someone judges others unfairly, then they can be judged in the same way.
When someone refuses to show others mercy, then society can refuse them mercy as well.
Pros and Cons:
This form of justice works well for keeping a governmental system running smoothly, but it gives no path for criminals to integrate back into society. It keeps people who have failed society cut off and shunned.
Retribution:
Now for the second part: retribution. There are two things that effect your engagement in society. The first is the social contracts that you follow, and the second is the values that you have, which come from the identity ethics. If a person breaks a social contract you can just end the contract, but if someone's values don't align with society's then society might try and end their values.
Which is a little problematic because no one actually has the power to change your values. Their only option is to create an incentive for adopting society’s values. Often through harsh punishments, and occasionally through rewards.
This form of justice doesn't have to apply to societies exclusively either, it can apply to an individual. If someone's values don't give you power, then retributive justice states that you have the right to make them change.
If someone does something you believe is wrong, then you can retaliate so that they would never think of doing that again. For instance the law of Moses recommends "an eye for an eye". However this form of justice does not mandate that the punishment you inflict is equal to the crime. You could deal back as great or small of a punishment as you have the power to inflict.
And this can apply to any of the identity ethics. If someone represents something you don't agree with, makes a decision you don't agree with, makes a judgement you don't agree with, or isn't showing you enough support, then you can retaliate with as much force as you see fit.
Results
So what is stopping someone from completely destroying their enemies? There are two things, first is the fear of being destroyed by someone else. The second is the consequences of breaking social contracts. Often societies will restrict the amount of violence that people can inflict on each other using social contracts.
As a result a society which values the identity ethics more, such as a dictatorship, will end up being run by the people who have the most power, and can thus carry out their version of justice unopposed. These people become the elite of society.
This is problematic for the rest of us because these leaders will have a biased sense of justice. There are things that they will enforce, with or without the rest of society's permission, that keep them in power. For example fascist regimes enforce racism.
A government which values the relationship ethics more will tend not to have an elite class, and will instead enforce laws by removing their ability to benefit from society. Some religious groups enforce rules like this, for instance members of my church, the church of Jesus Christ of latter-day saints, who sin can be given membership restrictions.
But this is not the best way for a government to be run. A better way is to help people to keep the law, and help them to become better people, and forgive them.
Forgiveness:
Mercy is forgiving a person's offenses. But mercy needs to be deserved, or it would rob justice.
If someone's values offend you, then they don't deserve your retribution, and you should forgive them unconditionally.
However, if someone has destroyed important relationships it is a safety concern to just let them continue their behavior. In order to forgive them another form of justice is needed, called restorative justice.
Restorative justice seeks to restore the relationships of both the victim and the criminal. It demands the criminal makes an apology that the victim is willing to accept.
There are five different ways people can apologize, they can stop their behavior, make restitution, request forgiveness, accept responsibility, and express regret. When you have adequately apologized in all these ways then there shouldn't be anything left in the way of restoring your relationships.
Part of showing mercy is giving a person chances to apologize, and helping them to make the necessary changes in their life to do so. This part of restorative justice often requires a mediator who is willing to work with the criminal and the victims to help them to heal.
If a person still can't forgive you after you have apologized, then retributive justice will have to take its course, even though you don't really deserve it anymore.