Grace is basically when God does something for us that we could not possibly do for ourselves. For example, the Bible says; For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast. Ephesians 2:8-9 It is God who saves us when we clearly cannot be saved by anything that we can do. It is out of His goodness and undeserved love, that he is working in our lives by His grace.
For sin shall not have dominion over you: for ye are not under the law, but under grace. Romans 6:14 We, as members of God's church are not under the law, but under God's grace. We are not to struggle in the flesh trying to keep God's laws. Instead, we are to believe God and trust Him to live out His life through us. We can't live "the Christian life", but God can, inside of us.