Atonement in Judaism is the process of causing a transgression to be forgiven or pardoned.
In Rabbinic Judaism, atonement is achieved through repentance, which can be followed by some combination of the following:
confession
restitution
the occurrence of Yom Kippur (the day itself, as distinct from the Temple service performed on it)
tribulations (unpleasant life experiences)
the experience of dying.
the carrying out of a sentence of lashes or execution imposed by an ordained court (not now in existence)
Temple service (not now in existence, e.g. bringing a sacrifice).
Which of these additions are required varies according to the severity of the sin, whether it was done willfully, in error, or under duress, whether it was against God alone or also against a fellow person, and whether the Temple service and ordained law courts are in existence or not. Repentance is needed in all cases of willful sin, and restitution is always required in the case of sin against a fellow person, unless the wronged party waives it.
According to Maimonides, the requirements for atonement of various sins between man and God are as follows:
The sentence of an ordained court (when available) can also substitute for Yom Kippur + tribulations + dying.
In Judaism, once a person has repented, they can be close to and beloved of God, even if their atonement is not yet complete.
The Mishnah states:
To a man who says, 'I will sin and repent, I will sin and repent', Yom Kippur brings no atonement. For sins against God, Yom Kippur brings atonement. For sins against one's fellow man, Yom Kippur brings no atonement until he has become reconciled with the fellow man he wronged.
According to Maimonides, in order to achieve true repentance the sinner must abandon their sin, remove it from their thoughts, and resolve in their heart never to repeat it, as it is said, “Let the wicked forsake his way and the man of iniquity his thoughts" (Isaiah 55:7). Likewise, they must regret the past, as it is said, "Surely after I turned I repented" (Jeremiah 31:18).