What is the definition of mercy?

Mercy, pity, and compassion are roughly synonymous. While most modern versions are very good at capturing the meaning of the original languages, the best way to get at the meaning of a word as it is used in Scripture is to read every instance of the word in Scripture. One can do this using a standard concordance or any of several online Bible programs or Logos Bible Software. These methods make it easy to find every instance of a word used in the Bible.
A few verses will serve to illustrate. The following verses highlight mercy as it relates to forgiveness or the withholding of punishment:
• Have mercy on me, O God, according to your unfailing love; according to your great compassion blot out my transgressions. (Psalm 51:1)
• You, Lord God Almighty, you who are the God of Israel, rouse yourself to punish all the nations; show no mercy to wicked traitors. (Psalm 59:5)
• Let the wicked forsake their ways and the unrighteous their thoughts. Let them turn to the Lord, and he will have mercy on them, and to our God, for he will freely pardon. (Isaiah 55:7)
• So he called to him, “Father Abraham, have pity on me and send Lazarus to dip the tip of his finger in water and cool my tongue, because I am in agony in this fire.” (Luke 16:24)
• But the tax collector stood at a distance. He would not even look up to heaven, but beat his breast and said, “God, have mercy on me, a sinner.” (Luke 18:13)
• Just as you who were at one time disobedient to God have now received mercy as a result of their disobedience, so they too have now become disobedient in order that they too may now receive mercy as a result of God’s mercy to you. For God has bound everyone over to disobedience so that he may have mercy on them all. (Romans 11:30–32)
• But for that very reason I was shown mercy so that in me, the worst of sinners, Christ Jesus might display his immense patience as an example for those who would believe in him and receive eternal life. (1 Timothy 1:16)
These verses highlight mercy as it relates to having compassion upon a person with a need:
• A Canaanite woman from that vicinity came to him, crying out, “Lord, Son of David, have mercy on me! My daughter is demon-possessed and suffering terribly.” (Matthew 15:22)
• “Lord, have mercy on my son,” he said. “He has seizures and is suffering greatly. He often falls into the fire or into the water.” (Matthew 17:15)
• Two blind men were sitting by the roadside, and when they heard that Jesus was going by, they shouted, “Lord, Son of David, have mercy on us!” (Matthew 20:30)
• As Jesus was getting into the boat, the man who had been demon-possessed begged to go with him. Jesus did not let him, but said, “Go home to your own people and tell them how much the Lord has done for you, and how he has had mercy on you.” (Mark 5:18–19)
• We have different gifts, according to the grace given to each of us. If your gift is prophesying, then prophesy in accordance with your faith; if it is serving, then serve; if it is teaching, then teach; if it is to encourage, then give encouragement; if it is giving, then give generously; if it is to lead, do it diligently; if it is to show mercy, do it cheerfully. (Romans 12:6–8)
In the Bible, mercy is extended to an offender in the form of forgiveness or to the suffering in the form of healing or other comfort. In any case, mercy can be characterized as compassionate treatment of those in distress. Whether the distress is caused by the guilt or penalty of sin or by a debilitating physical condition, mercy is there to help.
In common usage, mercy and grace are often used interchangeably. They do not mean the same thing, but they are integrally related and may be considered two sides of the same coin in salvation. When God saves a person, He extends both mercy and grace. Mercy is forgiving the sinner and withholding the punishment that is justly deserved. Grace is heaping undeserved blessings upon the sinner. In salvation, God does not show one without the other. In Christ, the believer experiences both mercy and grace.