Jewish law requires us to ask forgiveness from anyone whom we may have harmed, whether the harm was physical, financial, emotional, or social. We are also required to be gracious in granting forgiveness. The Talmud (Bava Kamma 8:7) states, "From where do we know that it is cruel to not forgive? For it says, "Abraham prayed to God and God healed Abimelech..." (Genesis 20:17).
Maimonides writes (Hil. Teshuvah 2:9-10):
Repentance and Yom Kippur only atone for sins between Man and God such as eating forbidden foods or engaging in forbidden sexual relations. Sins between one man and his fellow, such as striking, cursing, or stealing are never forgiven until one pays up his debt and appeases his fellow. Even if he returns the money he owes he must still ask for forgiveness. Even if he only spoke badly about him, he must appease and beseech until he is forgiven. If his fellow refuses to forgive him then he must bring a group of three of his fellows and go to him and ask him [for forgiveness]. If he still does not forgive him he must go to him a second and third time (with a different group of three people). If he still refuses to forgive him he may cease and the other is the sinner. If [however, the injured party] is his teacher (rebbe) he must go to him even a thousand times until he is forgiven.
It is forbidden to be cruel and difficult to appease, rather, a person must be quick to forgive and difficult to anger and when the sinner asks for forgiveness he should forgive him willingly and wholeheartedly....
In Shulchan Aruch (R' Yosef Karo, d.1575) in the laws of Yom Kippur (Orech Chaim 606:1) we find essentially the same thing. The Rema (R' Moshe Isserles, d.1572) adds that one may withhold forgiveness if it is for the benefit of the person asking. The Mishna Berurah (R' Yisrael Meir Kagan, d.1933) explains that it may be appropriate to withhold forgiveness to teach the supplicant not to take it lightly. The Rema also permits withholding forgiveness when someone spread false rumors about you, however the Mishna Berurah says that in such a case one should still forgive.
In the prayer titled Tefilah Zaka which many people recite before Kol Nidrei on the eve of Yom Kippur it says:
I extend complete forgiveness to everyone who has sinned against me, whether physically or monetarily, or spoke lashon hara (negative speech) about me or even false reports. And (I also forgive them) for any damages, whether on my body or my property, and for all sins between a man and his fellow except for money which I can claim in a court of law and except for someone who sins against me saying, "I will sin against him and he will forgive me". Except for these I grant complete forgiveness and no person should be punished on my account. And just as I forgive everyone so should You grant me favor in the eyes of all men that they should completely forgive me.
The complete prayer is printed in many machzorim (holiday prayer books).