The Meaning of Grace

What does Grace really mean? The difference between the old law (What was given to the Jews in the old testament) and the new covenant (what Christ did for us on the cross) is a lot like a marriage. When I was married there were certain vows I took. Those vows are made as a contract to hold together the marriage. However I can still harm my relationship with my wife while staying inside those contractual bounds. If I forget my anniversary and do not try to make amends, I am not breaking the contract, but I am hurting the relationship. Jesus came as a sacrifice so that a covenant could be made between God and man. In a sense what this means as Christians we are no longer under the law, as in the cold contract, but we are now in a covenant relationship with Christ. Meaning we work through our issues with Him and are not bound to the repercussions of breaking a contract. Which I think is why we are called the Bride of Christ. And it is also what is meant by Jesus is the fulfillment of all the law and the prophets because He gave us a relationship which was the ultimate goal of the law, but was never realized in the law.