Jesus said, "For if you forgive others their trespasses, your heavenly Father will also forgive you, but if you do not forgive others their trespasses, neither will your Father forgive your trespasses" (M@ 6:14-15).
That's a hard truth.
Like love, forgiveness is not a feeling- it is a choice.
I few chapters later in Matthew 18 (ok, actually 12), Jesus tells the story of a slave released from the burden of his debt. This man, once forgiven for his immense debt, acted without remorse toward someone who owed him very little. He forgot (it seems as though it was only a matter of minutes, or days) that he had greatly been forgiven of something huge, and turned around to make a fuss over what was really nothing.
As followers of Christ and therefor receivers of His grace, we should certainly extend the act of forgiveness to those around us (if we have been forgiven in the ultimate way, we should be able to forgive people for bad choices). Yet I challenge you to see it also in a different way: we should be compelled to show mercy and forgive based on the joy that it brings and life that it gives.
I've learned the importance of the process. When you need to apologize, do it genuinely- you need to really understand the hurt you have caused before truly seeking forgiveness. If you are on the other end, when someone looks to you for forgiveness, be gracious and mature and let them know they are forgiven (but only if they really really are). I challenge you to actually say the words "I forgive you." It will serve as a powerful reminder...