The primary definition of "mercy" is an expression of compassion, particularly toward an offender or someone inferior. In the Bible, God's treatment of sinners is often called merciful. Mercy sometimes applies to compassionate treatment toward people in distress.
While mercy and compassion are very linked, compassion usually refers to the emotion that arises when someone becomes aware of another person's suffering, while mercy refers to the actions that are a result of the initial emotion. In the Bible, however, mercy most often refers to not receiving something deserved, such as punishment. Grace in the Bible is closer to the standard definition of mercy; biblical grace refers to receiving something undeserved, such as favor.