Gemara

(Mishnah): Even though the damager pays the victim, he must ask the victim for forgiveness - "return his (Avraham's) wife to him (and he will pray for you)".

If he does not forgive, he is cruel - "Avraham prayed... and Hash-m cured Avimelech..."

Yoma 85b (Mishnah): Yom Kipur does not atone for sins Bein Adam l'Chavero until the sinner appeases the hurt party.

Once, Rav was reading in front of Rebbi. Three Chachamim entered at different times. Each time, Rav started again from the beginning. When R. Chanina entered, Rav 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: Why didn't R. Chanina forgive him? Rava taught that if one is Ma'avir Al Midosav (pardons others who mistreated him), all his Aveiros are overlooked!

Answer: R. Chanina saw in a dream that Rav would become Rosh Yeshiva. He refused to forgive him, so that Rav would to go to Bavel and teach Torah there.

Shabbos 149b (R. Yakov brah d'Vas Yakov): If someone is punished due to Ploni), Ploni does not enter Hash-m's Mechitzah.

It says "Gam Anosh la'Tzadik Lo Tov." "Lo Tov" is Ra, and it says "Ki Lo Kel Chofetz Rasha Atah Lo Yegurcha Ra" - Ra will not dwell in Hash-m's dwelling.

Yevamos 79a: David tried to appease the Giv'onim; they refused. David said 'there are three Simanim of Yisrael: they are merciful, have a sense of shame, and bestow Chesed.' The Giv'onim lack mercy, so they may not marry Yisre'elim.

Rishonim

The Rif and Rosh (8:16) bring our Mishnah.

Rambam (Hilchos Chovel u'Mazik 5:10): If Reuven injured Shimon, Shimon may not be cruel and not pardon. This is not the way of Yisrael. Rather, once Reuven requested forgiveness and supplicated once and twice and Shimon knows that he repented, he should forgive him. Whoever is quick to pardon is praiseworthy; Chachamim are pleased with him.

Iyun Yakov (Shabbos 149b): If someone is punished due to Ploni and Ploni desires this, Ploni does not enter Hash-m's Mechitzah.

Rambam (Hilchos Teshuvah 2:10): One may not be cruel and not be appeased. He should be easily appeased and hard to be angered. When the sinner asks forgiveness, one must forgive willingly with all his heart. Even if he greatly pained him and sinned against him, he may not avenge this or bear a grudge. This is the way of Yisrael.

Poskim

Shulchan Aruch (OC 606:1): Yom Kipur does not atone for Aveiros Bein Adam l'Chavero until he appeases him. This is even if he only irritated him with words. If Shimon is not appeased the first time, Reuven must return a second and third time, each time with three people. If he is still not appeased, Reuven is exempt. If one offended his Rebbi, he must ask many times until his Rebbi is appeased.

Rema: Shimon may not be cruel and not pardon, unless he intends for Reuven's benefit.

Question (Bach DH v'Da and Gra 5): We learn from R. Chanina that it is forbidden only if he intends to be cruel. If he has a reason, e.g. he will benefit from it, he may do so. The Rema permits if he intends for Reuven's benefit. He explains that R. Chanina wanted Rav to go to Bavel, lest Rav be burdened by needs of the Tzibur and be Batel from learning. One may not do so for his own benefit. What is the Rema's source? Rashi explains oppositely!

Answer (Taz 2): The Rema explains like Rashi. He infers that just like one may refuse to pardon if he seeks his own benefit, he may refuse if he seeks the sinner's benefit, to humble his uncircumcised heart.

Pri Chodosh: The Rema connotes that whenever he has a reason, be it his own or Shimon's benefit, this is not cruel and it is permitted.

Chachmas Shlomo: How can we learn from R. Chanina? Chachamim sting like a scorpion, i.e. they do not forgive (Perush ha'Mishnayos on Avos 2:10). Surely, they do not transgress! Rather, they are not obligated to forgive. How do we learn from Avraham that one must forgive? Rather, a Chacham must forgive worldly things (e.g. taking Sarah), but not spiritual matters (e.g. Gechazi ruined Elisha's Kidush Hash-m). The Gemara thought that R. Chanina refused to forgive a worldly matter (a lack of honor), and answered that it was a spiritual concern. Alternatively, Chachamim need not forgive, due to Kavod ha'Torah. This did not apply before Matan Torah, e.g. to Avraham. A Chacham must forgive another Chacham, for it belittles Torah if the offender must ask forgiveness many times. Berachos 28a supports the first explanation; all assumed that R. Yehoshua did not forgive R. Gamliel.

Mishnah Berurah (9): Even when he seeks to humble Reuven, he must forgive him in his heart, for he requested forgiveness.

Mishnah Berurah (8 and Sha'ar ha'Tziyon 8): If one forgives Shogeg or Mezid injustices, his analogous Aveiros will be forgiven; if not, they will not.

Magen Avraham (4): Likewise, one may refuse to pardon if he seeks his own benefit, like R. Chanina did.

Mishnah Berurah (10): He need not pardon if he fears lest this harm him.

Tosfos Yom ha'Kipurim (Yoma 87b DH veha'Rav): The Rema explains that R. Chanina intended that Rav become Rosh Yeshiva in Bavel to benefit Rav. Rashi's Perush does not answer why R. Chanina refused to forgive! It would be better for him to forgive, and then his Aveiros will be pardoned! Perhaps not pardoning (so R. Chanina can live longer) would cause R. Chanina to die, since his Aveiros will not be pardoned! Rather, R. Chanina pardoned Rav in his heart, but did not reveal this so Rav would go to Bavel for Rav's benefit.

Note: Rashi could likewise explain that R. Chanina pardoned in his heart, but did not reveal this so Rav would go to Bavel for R. Chanina's benefit!

Yam Shel Shlomo (Yevamos 8:27): Full atonement is only after Shimon prays for him. Reuven should request this; it is a great Mitzvah for Shimon to do so.

Rema: If Reuven was Motzi Shem Ra on Shimon, he need not forgive him.

Source (Darchei Moshe 2 and Beis Yosef CM 420:33 DH v'Chasuv): The Mordechai (Yoma 723), Semag, Hagahos Maimoniyos (Teshuvah 2:9) and Mahari Veil say that one need not pardon Motzi Shem Ra (Yerushalmi Bava Kama 8:7). Perhaps someone heard Reuven be Motzi Shem Ra, and did not hear that he was appeased. For other things, he should not be cruel.

SMA (422:6): Shem Ra assails his ancestors and seed for all generations.

Taz (3): The Mordechai connotes that Shimon is not cruel if he refuses to pardon, but if he pardons, it helps. This is why if Shimon died, Reuven goes to his grave to ask forgiveness.

Yam Shel Shlomo (Bava Kama 8:3): If one overcomes his Yetzer and totally forgives even this, even though it is difficult, it shows humility and Chasidus.

Kaf ha'Chayim (33): Even for Motzi Shem Ra, Reuven need not request forgiveness more than three times. Lev David disagrees.

Shulchan Aruch (CM 422:1): If Reuven injured Shimon, he may not be cruel and not pardon. This is not the way of Yisrael. Rather, once he asked forgiveness once and twice and Shimon knows that he repented, he should forgive him. Whoever is quick to pardon is praiseworthy; Chachamim are pleased with him.

SMA (5): In OC 606, we learn that the offender must request three times. The victim should forgive the first or second time.

Bach (Sof ha'Siman): Even though the offender must ask many times, one should forgive a friend the first time he asks, and a Talmid the second time.