Ooh, this could be a doozy of a conversation. Did you have a particular situation in mind when you posed this question? Or are you speaking more broadly, as a way to handle forgiveness throughout one's life?

Either way, I think there are two major factors in one's ability to forgive: Knowledge and willpower.

Knowledge that vengeance is an empty victory. Often, people choose vengeance as a way to help mitigate their emotional pain, and rarely does it solve the problem. For example, a murderer kills your friend, so you kill the murderer to avenge your friend. It won't bring your friend back, and you're still left with the emotional turmoil of losing your friend. On a broader level, a philosophy where vengeance is highly valued can lead to a violent society full of strife and misery. Take the death penalty - there's evidence to suggest that countries and states which use the death penalty have more violent crimes and murders than states that do not.

Willpower to ignore your base instincts. You need to be able to see the issue clearly so that you can work to solve the problem, even if that's merely stopping the cycle of violence. To see the issue clearly, often this means you need to be able to understand the motives of the one who committed the crime.

The Stoics have a lot to say on this subject, and I'd be happy to discuss their thoughts, if you'd like.