Mercy is not being given a punishment that we deserve. It is not being punished as our Sin deserves.
If I were a high school student and didn’t do my homework, I deserve to be given a detention. The teacher would show mercy on me if they decided not to give me the detention.
Grace is loving someone despite the wrong they have done. In short Grace is the undeserved love of God.
In the 2012 film Les Miserables, Valjean, a convict who has broken his parole, is taken in by a Bishop. Valjean decides to run off with the Bishop’s silverware. He is caught by a policeman and brought back to the Bishop. The Bishop shows mercy on Valjean by saying to the policeman that he had given the silver to Valjean. This act of mercy ensures that Valjean does not get punished. The Bishop then shows Grace to Valjean by letting him keep the silver, telling him to use it to make him a better man. The Bishop has shown Valjean love which he doesn’t deserve.
Through Jesus, God our Father uses mercy and grace to atone for our sins and to bring us salvation.
For more on sin and atonement see: Sin and Atonement, Atonement Part II
For more on salvation see: Salvation
For more Theological examples using Les Miserables, see: The Theology of Love, Part IV