What the Bible says
“Think not that I am come to send peace on earth: I came not to send peace, but a sword.”
Matthew 10:34 (KJV)
“Suppose ye that I am come to give peace on earth? I tell you, Nay; but rather division.”
Luke 12:51 (KJV)
“Peace I leave with you, my peace I give unto you: not as the world giveth, give I unto you.”
John 14:27 (KJV)
Luke says 'division'
Luke records the same teaching with the word division. The next verses in Matthew explain it: a man set at variance against his father, and a person's foes being those of his own household. Jesus was preparing his disciples for the cost of following him, not handing them weapons.
Not a call to violence
When Peter swung a real sword in Gethsemane, Jesus told him to put it away, for all who take the sword shall perish with the sword (Matthew 26:52). Elsewhere God's word is itself called a sharp two-edged sword (Hebrews 4:12). Truth divides between what is of God and what is not.
Two kinds of peace
Jesus does not bring a peace that keeps everyone comfortable by ignoring the truth. He brings peace with God, and a peace in the heart that holds even when others turn away. That is why the one who spoke of a sword could also say, let not your heart be troubled.
Common questions
Did Jesus approve of violence?
No. He told Peter to put away his sword (Matthew 26:52). The 'sword' in Matthew 10:34 is the division that loyalty to him can cause.
Isn't Jesus the Prince of Peace?
Yes. He gives peace with God and peace of heart (John 14:27), but that peace can bring conflict with a world that rejects him.