What is the punishment of God?

Answer 1

Punishments from God varied. Some were minor, especially if a person's sin (or transgression) was small and not intentional; those who committed these offences could bring a guilt or sin offering to the high priest and apologize to God for the sin. But some punishments were more severe. For example, in the Hebrew Bible (Old Testament), we are told that Aaron had two sons who were supposed to be of priestly lineage (Aaron was a high priest and the brother of Moses). But when they disobeyed God and offered sacred fire that they had not been asked to offer (some sources say they offered worship to pagan gods, rather than to the One God), they were immediately killed as punishment for their disobedience. Moses, as great a leader as he was, lost his temper with the Israelites and angrily struck a rock, disobeying God's commandment to speak gently so that water would flow from it. As punishment, Moses was not allowed to enter the Promised Land.

It should be noted that the Bible is filled with stories of people who sinned but were not killed; some were punished with a tragedy (King David committed adultery, and was punished with the death of his little son); or given an illness (Miriam questioned Moses marrying a Cushite woman, and Miriam was punished with a skin disease that is often incorrectly translated as leprosy); or stigmatized in their community until the person repented and publicly apologized for having sinned. The emphasis of most Bible stories was that the Lord was a forgiving God: He demanded faithfulness and dedication, but if people sinned, they needed to sincerely repent. Those who truly showed remorse and repentance could earn His forgiveness, rather than His punishment.

Answer:

This can be split into two questions.
1) Punishments in this world
2) Punishments in the afterlife.
1) The Torah gives authority to the Sanhedrin (Jewish court) to punish certain types of crimes. They are bound by 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.
Since shortly before the destruction of the Second Temple, capital cases were no longer judged, 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) 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, neither 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. See for example the Talmud, Rosh Hashanah 17a, for a few details.
It is important to note that Judaism is a rare case in that it does not condemn non-adherents to eternal damnation. Worthy non-Jews are able to merit 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.