There are not very many ideas routinely characterized as Christian (regardless of their actual origin) that I like, but forgiveness is one of them. Granted, I find all the stuff about divine forgiveness, sin, salvation, heaven, and the like to be laughably absurd. These notions are little more than the most primitive forms of wish fulfillment, and we would do well to abandon them. However, the idea that we should strive to forgive others who have wronged us does hold some appeal.
Christians are sometimes criticized for pushing forgiveness to unrealistic extremes (e.g., turning the other cheek). That may be deserved. I think we can all imagine scenarios where forgiving someone might be almost impossible or even contraindicated. And if forgiveness is about the other person, then we can find countless examples of persons or organizations who do not deserve forgiveness in the least.
But forgiveness, at least the sort I'm talking about here, is not really about the other person at all. It is about us. As something we can do for ourselves, at least in some circumstances, striving to forgive others is a decent idea with benefits.
When we hold grudges, resentments, or hatred, we invest our own emotional resources in something that typically ends up being toxic for us. Forgiveness, when possible, relieves this burden and is good for our physical and emotional well-being.
Forgiveness is not the same thing as forgetting, and it certainly does not absolve anyone of guilt, blame, or responsibility (sorry President Obama, you can't have this as an excuse to ignore Dick Cheney's war crimes). I view it primarily as the process of coming to terms with pain another has inflicted. It is a way of reducing the emotional hold the injustice we suffered continues to have on us.