In most religions forgiving each other or by God is very common but is in Buddhism forgiving something wiht any value? If so what does it really mean, what does it effect and how is it done?
Hatred is, indeed, never appeased by hatred in this world. It is appeased only by loving-kindness. This is an ancient law.
In Buddhism forgiveness is between one another than by an external agency. Having said this no external agency is behind Karma. It is only your tainted or pure mind. And once done the results follow.
All mental phenomena have mind as their forerunner; they have mind as their chief; they are mind-made. If one speaks or acts with an evil mind, 'dukkha' follows him just as the wheel follows the hoofprint of the ox that draws the cart.
... similarly for good deeds good results follow ...
All mental phenomena have mind as their forerunner; they have mind as their chief; they are mind-made. If one speaks or acts with a pure mind, happiness (sukha) follows him like a shadow that never leaves him.
Following highlights no external agency can affect Karma than the mind itself:
This is explained in the Pathama Mahanama Sutta, by the simile of an earthern pot filled partly with gravels and stones and partly with fat and butter. By throwing this pot into water and smashing it with a stick, it will be seen that gravels and stones quickly sink to the bottom while fat and butter rise to the surface of the water. Likewise, when a person who has established himself in the five wholesome dhammas of faith, conduct, learning, charity and insight dies, his body remains to get decomposed but his extremely purified mental continuum continues in higher states of existence as birth-linking consciousness, patisandhi citta.
In the concluding suttas are expositions on the Middle Path, the Four Noble Truths and the Noble Path of Eight Constituents.
Source: Guide to Tipitaka - 6. Samyutta Nikaya by Professor Ko Lay
A purpose of Buddism is to be without 'defilements' -- Wikipedia introduces Kleshas (Buddhism) as,
Kleshas (Sanskrit: kleśa; Pali: kilesa; Standard Tibetan: nyon mongs), in Buddhism, are mental states that cloud the mind and manifest in unwholesome actions. Kleshas include states of mind such as anxiety, fear, anger, jealousy, desire, depression, etc. Contemporary translators use a variety of English words to translate the term kleshas, such as: afflictions, defilements, destructive emotions, disturbing emotions, negative emotions, mind poisons, etc.
I would add verse 3 and 4 of the Dhammapada to the verses which Suminda's answer already quoted:
"He abused me, he ill-treated me, he got the better of me, he stole my belongings;"... the enmity of those harbouring such thoughts cannot be appeased.
"He abused me, he ill-treated me, he got the better of me, he stole my belongings;"... the enmity of those not harbouring such thoughts can be appeased.
If you feel hurt by someone and will not "forgive" them, then you would retain feeling hurt and enmity.
So perhaps you should take it for granted that "not harbouring such thoughts" is desireable.
One more thing is that, further to the above, I think it's difficult to hurt an enlightened person.
So assuming that you will forgive people, perhaps the process of forgiveness (or "how it is done", as you asked) might be informative (enlightening): for example, you might learn what kind of clinging, clinging to what, allowed you to feel hurt in the first place; or you might learn further compassion for whoever was in a situation where they hurt you.
Lastly, some people say that the Vinaya isn't interesting to (doesn't apply to) lay people, but I think it's interesting because it details rules about how to live together in society.
I guess that according to the Vinaya, "forgiveness" isn't entirely one-sided i.e. the "injured" person isn't the only one who participates in the forgiveness process ... for example the wrong-doer too is also supposed to confess what they did wrong; and a wrong-doer who won't cooperate might be banished.
P.S., sometimes people want to know how reliable a statement is.
- I think the first statement above (which quotes the Dhammapada) is fairly reliable.
- The second (that you might learn something from the process of forgiving someone, i.e. learn what you did that allowed you to feel hurt in the first place) is my opinion.
- The third (about forgiveness and confession within the Vinaya) is not reliable, and is based on my superficial reading of the rules and not based on actual experience.