If you are forgiving, you are willing to let go of anger when someone has hurt you or done something wrong. You do not keep the hurt forever, and you do not keep looking for a way to pay them back. People use it about a person (“She is very forgiving”) and about a look or a way of being (“a forgiving smile”). Though actually, it does not mean you think the other person was right — only that you will not hold the wrong against them. English also stretches the word to things that are not people: a “forgiving” fabric still looks all right if it is a bit creased; a “forgiving” tool still works even if you do not use it perfectly. In that looser use, the thing “forgives” your small mistakes. The word is kind and everyday, not stiff. You will also see forgiving as the -ing form of the verb forgive (“She is forgiving him right now”); the sense here is the adjective: a person (or thing) that is ready to forgive.