Buddhists believe strongly that law is required to PROTECT society from harm. As justice will always be carried out by the law of karma the most important aspect of punishment will always be PROTECTION. Buddhists are not in favour of RETRIBUTION as they do not wish harm to anyone (ahimsa) and teach kindness (metta) and compassion (karuna) over all things. If we can help to REFORM a person so that they could change their behaviour for the better this would be the best way and would help them to keep the FIVE MORAL PRECEPTS.
Lastly, REPARATION might help a person to stave off the effects of bad karma. If you can willingly pay for the wrongs you have done then you will not have to be re-paid by karma in this life or the next.
Apart from RETRIBUTION Christians agree with most of the main aims of punishment. Obviously REFORM is key as Christianity believes in FORGIVENESS and turning the other cheek (see Matt 5:39). Also important would be removing the causes of crime which might include social and environmental causes. If we can create better living environments; give people better education and opportunity, then they are more likely to make good choices.
REPENTANCE is key to the idea of Christian forgiveness. People must be sorry for what they have done and make an attempt to live their life in a better way. We do not forgive people who are not sorry for what they have done or intend to do the same again. REPARATION can thus be important in helping people to make up for the wrongs they have done.
Like Buddhists, Hindus believe in the inescapable law of karma. A person who is criminal in this life will be punished with movement down the scale of samsara in their next life. In other words they may come back as an animal or woman as a punishment for wrongdoing now. As karma may move a person down through the varnas (castes) as punishment for wrongdoing this was assumed to provide a strong DETERRENT for members of the higher varnas (kshatriya (warrior) and brahmin (priests)) so that they would not commit crimes and risk moving down into the vaishya (merchant) or shudra (peasant) classes.
There is also an element of REPARATION linked to karma, if you can willingly make up for the wrong that you have done then this might mitigate (make better) the negative effects on karma. For this reason a death penalty was often offered to murderers so they could redress the balance of harm done before (hopefully) being reborn into the same varna and another chance to live life again with a clean slate. Thus RETRIBUTION is also important. If I can pay by having the same done to me as I did to another then I can rebalance my karma and move on with my life (or next life).
DETERRENCE is key to the aims of punishment in Islam, as is RETRIBUTION. Lex Talionis (the law of retribution) is still key in Islam and many Shari’ah punishments focus on public and humiliating displays. Women, for example, can be buried up the waist in a public square or stadium and stoned to death by the crowds for adultery. Persistent thieves can have their hands cut off; which not only has the PROTECTIVE effect of preventing them stealing again (!) but acts as a very public DETERRENT showing others what can happen to them for theft. REPENTANCE and REFORM are largely left to an individual and their own relationship with Allah.
As the Law in Judaism is the word of God all attempts for REPENTANCE are between an individual and God. Only God can grant FORGIVENESS, as it is His Law that has been broken (This is why the Pharisees were so upset by Jesus’ claims to be able to forgive sins). Punishments for breaking these laws are set out quite clearly in Torah. Some of these rules are REPARATIVE (such as being ordered to pay the value of an animal that falls into a pit you have left uncovered (Exodus 21:33)). Others are aimed at deterrence, such as the prohibitions on sexual immorality resulting in exile (Leviticus 18:29) or the commandment that adulterers are to put to death in Leviticus 20:10.
In Judaism there is also the annual Day of Atonement as prescribed in Leviticus 16. Jews must atone (apologise and make amends for) the sins of the previous year before starting the new year afresh. This will involve confession of their sins, REPENTANCE of them, prayer, fasting (going without food for the day) and often giving money to charity, possibly as a form of REPARATION.