The Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204) writes in his Hilchot Teshuva (2:9; listen to audio for precise citation) that a person earns atonement on Yom Kippur only for sins committed solely against God, such as eating forbidden foods or engaging in illicit relationships. Sins committed against one's fellow, however, such as stealing, causing damage to somebody's body or property, or even verbally insulting another, cannot be atoned through Yom Kippur alone. The perpetrator must also make whatever restitution payments are required by Halacha, and ask the victim for his forgiveness. Only by receiving the victim's forgiveness can the perpetrator earn atonement for his wrongdoing.
The Rambam adds that if the victim refuses to grant forgiveness, then the sinner should assemble three colleagues and approach the victim again, in an effort to convince him to forgive. If after three requests the victim still refuses to forgive, then it is he – rather than the perpetrator – who is deemed the sinner.
In the next passage (2:10), the Rambam emphasizes the importance of responding favorably and forgivingly to those who have caused one harm and now seek reconciliation. Regardless of the crime, he writes, one should accept the apology and forgive unhesitatingly. The Rambam writes that Jews are characteristically slow to anger and quick to forgive, whereas the pagans would customarily grow angry at the slightest provocation and refuse to grant forgiveness. We are obliged to follow the time-honored Jewish tradition to graciously forgive those who seek reconciliation after having committed a wrongful act.
If a person committed an offense against his fellow, and the victim passes away before the perpetrator asks for forgiveness, then the perpetrator should assemble ten men at the victim's grave and confess his sin (Rambam, ibid. Halacha 11). If the perpetrator owes a compensatory payment to the victim, he must pay the sum to the victim's inheritors. If he is not aware of any inheritors, then he should pay the sum to the local Bet Din (Rabbinical court) and confess his sin in the court's presence.
Summary: Yom Kippur cannot atone for sins committed against one's fellow unless he pays the required restitution payments and asks the victim for forgiveness. The victim should respond favorably and forgive. If the victim has passed away, one should assemble ten men by his grave and confess his sin in their presence.
