This is a good question. Let me start with grace. A good definition of grace is unmerited favor or undeserved favor. It is also one of the foundational words and concepts of Christianity. It is the favor that God gives to his people even though they do not deserve it.
Forgiveness, in light of the above discussion, can be seen as an outflow of grace. In other words, to forgive someone is to acknowledge that a wrong has been committed, but you do not hold that against a person.
In light of these points, here are some characteristics of grace and forgiveness: patience, kindness, mercy, and most of love. 1 Corinthians 13 in the New Testament is a great chapter to see of these characteristics.
"Grace" in a secular context refers to the manner in which one carries oneself, particularly in public. It denotes a certain refinement and sophistication in one's conduct. In a religious context, it has a considerably different meaning. Grace in a biblical context refers to the considerations bestowed upon sinners by God, as though assuring those who sin that they remain among God's children.
"Forgiveness," in contrast, refers to the act of responding to one's sins or errors with magnanimity and acceptance. This is not the same as acquiescing in inappropriate conduct, but it does denote a certain absolution towards those who sin or error. To forgive someone is to accept his or her apology as sincere and agree to put the matter behind oneself. In the religious context, grace and forgiveness are entirely compatible concepts, but forgiveness is the more human trait, as grace, in the biblical sense, is the prerogative of God only.