What are the key beliefs about punishment in Judaism?

Answer 1

Punishment is meant to educate and improve. Where the court cannot intervene, the matter is left to God.

The question can be split into two questions.

1) Punishments in this world

The Torah gives authority to the Sanhedrin (Jewish court) to punish certain types of crimes. They must follow procedure, and have two witnesses, and the perpetrator must be warned before he sins; otherwise he is let go. Also, the Sanhedrin must make a concerted effort to exonerate the defendant. Their role is not intended to be one of vengeance.

Shortly before the Second Destruction, the Sanhedrin ceased dealing with capital cases, because the onset of the Exile made it impossible to maintain the level of scholarship needed for the procedures of the Sanhedrin. Since then, the power of Jewish courts has diminished and is limited to imposing fines, if anything.

2) Punishments in the afterlife.

Does Judaism believe in the existence of hell? The name and details are probably different than in other religions, but the short answer is yes. "And they shall go forth, and look upon the carcasses of the men that have rebelled against Me; for their worm shall not die, nor shall their fire be quenched" (Isaiah 66:24). One of the central beliefs of Judaism is that one's soul continues to exist after death and is treated in accordance with the person's actions while he/she was alive. All outstanding accounts are settled after this life. Some cases illustrate this point, since this entire world wouldn't be enough to reward a Moses or punish a Hitler.

The afterlife is discussed at length in the Talmud, though the exact nature of the rewards and punishments is generally not given since it would be like describing the colors of a sunrise to someone who was blind from birth.

It is important to note that Judaism is a rare case in that it does not condemn people of other religions to eternal damnation. Worthy non-Jews are able to earn the rewards in the afterlife without conversion.

Another important point is the efficacy of repentance, which can change one's spiritual "balance" relatively easily.

Answer 2

This can be split into two questions.

• 1) Punishments in this world
  

The Torah gives authority to the Sanhedrin (Jewish court) to punish certain types of crimes. They are bound by careful procedure, they must have two witnesses, and the perpetrator must be warned before he sins; otherwise he is not punished by the Sanhedrin. Also, the Sanhedrin is enjoined to make a concerted effort to seek evidence that could exonerate the defendant. This means that they must be balanced and circumspect, no matter what the crime. Their role is not intended to be one of vengeance.
Shortly before the destruction of the Second Temple, the Sanhedrin ceased dealing with capital cases, because the increased persecutions of the Romans, and the onset of the present Exile, made it impossible to fully maintain the level of scholarship and expertise needed for the Torah-procedures of the Sanhedrin. Since then, the power of Jewish courts has been greatly diminished and is now limited to imposing fines, if anything.

• 2) Punishments in the afterlife.
  

Does Judaism believe in the existence of hell? The name and details are probably different than in other religions, but the short answer is yes. "And they shall go forth, and look upon the carcasses of the men that have rebelled against Me; for their worm shall not die, nor shall their fire be quenched" (Isaiah 66:24). One of the central beliefs of Judaism, as codified by Rambam (Maimonides, 1135-1204), is that the soul continues to exist after death and is treated in accordance with the person's actions while he/she was alive. All outstanding accounts are settled after this life. Some cases illustrate this point, since this entire world wouldn't be enough to reward a Moses or punish a Hitler.
The afterlife is discussed at length in the Talmud, though the exact nature of the rewards and punishments is generally not given since it would be like describing the colors of a sunrise to someone who was blind from birth.
It is important to note that Judaism is a rare case in that it does not condemn people of other religions to eternal damnation. Worthy non-Jews are able to earn the rewards in the afterlife without conversion.
Another important point is the efficacy of repentance, which can change one's spiritual "balance" relatively easily. See for example Deuteronomy 30:1-10 for one of the Tanakh's many passages on this topic.