sins, and what are grave sins? Grave sins are those for which one incurs the death penalty by the court or kareis (premature death). False and vain oaths are also considered grave sins, although they are not punished by kareis (premature death). Light sins are the violation of the other prohibitions and (the failure to perform) positive commandments that are not punishable by kareis (premature death).
[1:3] Now that the Beis Hamikdash (Holy Temple) is not standing, and there is no sacrificial altar for atonement, we are only left with repentance10. Through repentance, God forgives our sins, and no mention is made of these sins even if one sinned one's whole life and repented only in his final moments. As it is written, "The wickedness of the wicked will not cause him to stumble when be turns back from his wickedness." (Ychezkiel 33:12).
The day of Yom Kippur11 in and of itself atones for those who repent, as it is written12, "This day will bring about atonement for you." (Leviticus 16:30).
[1:4] Although repentance amends one's sins, and the day of Yom Kippur itself brings about atonement, (there are different degrees of sins, and accordingly, different stages of atonement). Some sins can be rectified at once, while others need the passage of time. What are these categories?
10 One who willfully refuses to repent a sin cannot be forgiven before death even if he suffered by the hand of God or was executed by Rabbinical court. After death his punishment is completed in order to cleanse his soul. A sin is forgiven when one asks forgiveness, if not he must be cleansed of sin. Punishment cleanses sin and does not forgive sin. If one repents a severe sin, he is partially forgiven and the punishment completes cleansing his soul.
After the destruction of the Holy Temple, there is no Azazel ceremony, one can only pray to God to have mercy on him and forgive the sins that he is not aware of. This is prayer and not penitence. God will forgive, and depending on the circumstances he may also punish him to cleanse his soul.
If he was executed by a Rabbinical court without repenting, his execution can help to change a decree that banished him from the World to Come, to meriting the World to Come through the righteous in the World to Come, after he is punished after death to cleanse his soul. So is the ruling with a city in Israel that rebels and worships idols. The decree on the inhabitants of that city is that they will not enter the World to Come. However, if the Rabbinical court sent in the military to punish them, their death by a ruling of a Rabbinical court will change their decree and they will enter the World to Come after their souls are punished after death.
If, however, the idolatrous city was destroyed by a foreign enemy and not by a ruling of a Rabbinical court, this type of death cleanses their souls and they forgo punishment after death. Nevertheless, they lack mitzvos to merit them in the World to Come, and can only merit the World to Come through the merits of the righteous in the World to Come.
11 The holiness of the day of Yom Kippur torments the soul if it sinned, and added to this is the fasting and other abstentions of Yom kippur. There are sins that are pending Yom Kippur for forgiveness after repentance because that day God forgives sins.
12 After the destruction of the Holy Temple, what does one do to be forgiven for sins that he does not know about and cannot repent? One should pray for mercy to be forgiven for these sins as is printed in the Yom Kippur prayer books. He should study Torah and halacha in order to know what is sin. He should teach Torah to others. He should give charity and help others. He should pray on the Jewish fast days to be forgiven for sins that are not known to him.