Gemara

85b - Mishnah: Yom Kipur does not atone for sins Bein Adam l'Chavero until the sinner appeases the hurt party;

R. Elazar ben Azaryah says, "Mi'Kol Chatoseichem Lifnei Hash-m Titharu" - Yom Kipur atones for Bein Adam la'Makom. It does not atone for Bein Adam l'Chavero until the sinner appeases the one he hurt.

R. Yitzchak: If Reuven irritated Shimon even with words, he must appease him. "Beni Im Oravta l'Re'echa Tokata la'Zar Kapecha Nokashta b'Imrei Ficha" - if you (were a guarantor and) owe money, pay it. If you sinned with words, bring friends to ask forgiveness.

Rav Chisda: He must come with three friends three times - "Yashor Al Anashim va'Yomer Chotosi (1) v'Yashar He'eveisi (2) v'Lo Shavah Li (3)".

If Shimon died, Reuven brings 10 people to the grave and says 'I sinned against Hash-m and against Shimon.'

R. Yosi b'Rebbi Chanina: One should not ask forgiveness from his colleague more than three times - "Ana Sa Na...v'Atah Sa Na" ('Na' is a request for forgiveness).

Once, Rav was reading a Parashah in front of Rebbi. When R. Chiya entered, Rav started again from the beginning, and again when Bar Kapara entered and again when R. Shimon bar Rebbi entered. When R. Chanina entered, Rav said that this is too much. He refused to start again. R. Chanina was offended. For 13 years Rav went to him on Erev Yom Kipur to appease him; R. Chanina was not appeased.

Question: But R. Yosi b'Rebbi Chanina taught that one should not ask forgiveness more than three times!

Answer: Rav is different.

Question: Why did R. Chanina refuse to forgive him? Rava taught that if one is Ma'avir Al Midosav (does not punish others who acted improperly towards him), all his transgressions are overlooked!

Answer: R. Chanina saw in a dream that Rav would become Rosh Yeshiva. Therefore, he refused to forgive him, so Rav would to go to Bavel and teach Torah there (and R. Chanina could be Rosh Yeshiva for many years in Eretz Yisrael).

Rishonim

The Rif and Rosh (8:17) bring the above teachings (in the name of different Chachamim). They also say that one must ask forgiveness from his Rebbi until he is appeased.

Rambam (Hilchos Teshuvah 2:9): Yom Kipur atones only for sins Bein Adam la'Makom, such as Isurei Achilah or Bi'ah. It does not atone for sins Bein Adam l'Chavero such as theft or cursing until he gives what he owes, appeases him and asks forgiveness. Even if Reuven sinned against Shimon through words, he must appease Shimon until he forgives. If Shimon is not appeased, Reuven brings three of Shimon's friends and requests forgiveness. If he refuses, Reuven does so a second and third time. If Shimon is still not appeased, Reuven may go away; now Shimon is sinning. If Reuven offended his Rebbi he must ask forgiveness even 1000 times until he appeases him.

Question (Lechem Mishneh): Why did Rav try only 13 times to appease R. Chanina?

Answer (Lechem Mishneh): Rav saw that R. Chanina did not want to be appeased, so there was no need to try.

Rambam (ibid. 11): If Shimon died before Reuven ask forgiveness, Reuven brings 10 people to the grave and says 'I sinned against Hash-m and against Shimon; I did such and such...

Rosh (8:24): On Erev Yom Kipur one must appease anyone who might bear a grudge against him. We learn from the episodes of Rav with R. Chanina and the butcher. Pirkei d'R. Eliezer relates that on Yom Kipur the Satan says that Yisrael are like angels - they are free of sin, there is Shalom among Yisrael...

Kaf ha'Chayim (OC 606:15): If there is enmity among Yisrael this enables the Satan to prosecute (Levush).

Poskim

Shulchan Aruch (OC 606:1): If Reuven sinned against Shimon, even through words, Yom Kipur does not Mechaper for him unless he appeases Shimon.

Magen Avraham (Reish ha'Siman): The Bach says that Reuven must tell Shimon how he sinned against him. It seems that he need not if it will be embarrassing to Shimon.

Kaf ha'Chayim: Ben Chaviv is unhappy with people who send an intermediary to appease. The Torah wants the sinner himself to do so. However, the Pri Chodosh allows this if it is difficult to go himself, or if he is more likely to appease Shimon through someone else.

Shulchan Aruch (ibid.): If Shimon is not appeased the first time, Reuven must return a second and third time, each time with three friends. If Shimon is still not appeased, Reuven is exempt.

Magen Avraham (1,2): Each time Reuven should try to appease in a different way. He may try to appease more than this, if it will not be a disgrace to Torah.

Kaf ha'Chayim (17,18): He need not take different people each time. The Shulchan Aruch connotes that even the first time he goes with three friends. The Rambam connotes that the first time he goes alone, and afterwards three times with three friends.

Rema: Afterwards, Reuven should say in front of 10 people that he requested forgiveness from Shimon.

Kaf ha'Chayim (23,24): The Gra (DH Mihu) explains according to a Medrash that Hash-m will have mercy on Reuven because Reuven lowered himself. The Mishnah Berurah (6) explains that it is so people will not suspect Reuven did not ask forgiveness. According to this, it is not necessary if people do not know that he sinned against him.

Shulchan Aruch (ibid.): If Reuven offended his Rebbi he must ask forgiveness many times until his Rebbi is appeased.

Source #1 (Beis Yosef DH u'Mah she'Chosav v'Hani): The Rif, Rambam and Rosh learn this from Rav, who asked forgiveness from R. Chanina 13 times. R. Chanina became Rosh Yeshiva after Rebbi, therefore he was considered Rav's Rebbi. They explain the Gemara's answer (regarding why Rav requested more than three times) to say that one's Rav is different, i.e. one must ask forgiveness from his Rebbi without limit. This is unlike Rashi, who says that Rav was stringent to ask more than is necessary.

Note: Shabbos 59b says that Rav was already in Bavel when R. Efes died and R. Chanina became Rosh Yeshiva! Perhaps Rav was stringent to consider himself a Talmid because Rebbi had said that R. Chanina would be Rosh Yeshiva. This could also explain why Rav went only 13 times.

Source #2 (Taz 1): They learn from R. Yosi b'Rebbi Chanina, who said that one should not ask forgiveness from his colleague more than three times. The limit does not apply to one's Rebbi. They agree that Rav went beyond the letter of the law. R. Chanina was not Rav's Rebbi, even though he was Rosh Yeshiva.

Magen Avraham (3): This stringency is for anyone from whom Reuven heard Divrei Torah, even if he is not his primary Rebbi.

Shulchan Aruch (ibid. 2): If Shimon died, Reuven brings 10 people to the grave and says 'I sinned against Hash-m and against Shimon.

Magen Avraham (7): If Reuven lives more than three Parsa'os away, he may send a Shali'ach. If Reuven spoke badly about Shimon after Shimon died, he may ask forgiveness in that place.

Rebuttal (Kaf ha'Chayim 41): The Rema (CM 420:38, brought below) requires going to the grave even if he spoke after Shimon died. Yam Shel Shlomo says that a son may not pardon the honor of his deceased father, the sinner must ask forgiveness (himself or through a Shali'ach) from the father at the grave.

Rema: The custom is to ask forgiveness on Erev Yom Kipur.

Mishnah Berurah (1): Even though the obligation to appease applies the entire year, one who does not have the opportunity may wait. However, it is obligatory on Erev Yom Kipur in order that Yom Kipur will atone for his sins.

Kaf ha'Chayim (7): If Shimon does not know that Reuven owes money to him, Reuven must tell to him everything that he did and ask a qualified Rav what he owes. She'alas Shmuel is unsure whether or not Reuven must inform and appease Shimon (or if it suffices to return the money) in a case when Shimon was not pained, e.g. Shimon was unknowingly overcharged. It seems that he is obligated, for the Yerushalmi permits to bear a grudge against one who prevented him from using his money (until he is appeased).

Kaf ha'Chayim (44): Everyone should request forgiveness from his parents on Erev Yom Kipur. Most people are deficient in honoring parents every day. Even if the child did not request, the parent should say that he forgives totally. A husband should forgive his wife for spending money against his wishes. A Talmid should ask forgiveness from his Rebbi.

Rema (CM 420:38): If Reuven spoke about someone deceased, if he lives near the grave he must go to ask forgiveness in that place. If he is far away he may send a Shali'ach.