"You have heard that it was said, 'An eye for an eye and a tooth for a tooth.' But I tell you not to resist an evil person. But whoever slaps you on your right cheek, turn the other to him also. If anyone wants to sue you and take away your tunic, let him have your cloak also. And whoever compels you to go one mile, go with him two. Give to him who asks you, and from him who wants to borrow from you do not turn away" (Matthew 5:38-42).
The phrase "An eye for an eye and a tooth for a tooth" appears three times in the Old Testament (Exodus 21:22-25; Leviticus 24:17-22; and Deuteronomy 19:16-21). In each case, these were sentencing guidelines to judges after someone received a guilty verdict for particular crimes. From Jesus' comments, we realize that the Jews had taken these out of context and applied them as justification for revenge. If you hit me, I get to hit you back. The problem is the person is serving as his own prosecutor, judge, and jury -- and he's biased! These laws were intended to be used by those in authority who are upholding the law, not by people who think they are a law unto themselves.
The simple answer is that personal revenge is wrong. "If it is possible, as much as depends on you, live peaceably with all men. Beloved, do not avenge yourselves, but rather give place to wrath; for it is written, "Vengeance is Mine, I will repay," says the Lord" (Romans 12:18-19). But a lack of personal vengeance does not imply that evil cannot be prosecuted. "For rulers are not a terror to good works, but to evil. Do you want to be unafraid of the authority? Do what is good, and you will have praise from the same. For he is God's minister to you for good. But if you do evil, be afraid; for he does not bear the sword in vain; for he is God's minister, an avenger to execute wrath on him who practices evil" (Romans 13:3-4).
If you understand the necessity of having police to handle the evil people in the world, then you must also acknowledge that there are evil governments in the world as well because people are involved. A government has the right to protect its people from the evils of others. These should not be acts of personal retaliation, but an exercise of the authority God gave to governments.
It's the Luke quote that really baffles me. It says to love our enemies and do good to those who hate us, yet I can't see how killing people in war is good or loving: It seems contradictory to Paul's writing about what love is. Am I misapplying the passages here?
"But I say to you who hear: Love your enemies, do good to those who hate you, bless those who curse you, and pray for those who spitefully use you. To him who strikes you on the one cheek, offer the other also. And from him who takes away your cloak, do not withhold your tunic either. Give to everyone who asks of you. And from him who takes away your goods do not ask them back. And just as you want men to do to you, you also do to them likewise" (Luke 6:27-31).
When a police officer is enforcing the laws, he doesn't do so because he sees all law-breakers as his personal enemy. He will naturally hate the murderer, the abuser, etc. for the harm that he does, but it isn't a matter of personal vengeance.
What Jesus is talking about are times when it is personal -- when someone attacks you because of who you are. In those cases, you handle the situation by countering with doing good. Paul's writings about love are between individuals. This doesn't work in situations between a police officer and, say, a murderer. The officer isn't going to overcome by giving the murderer flowers. That is because, in any society, there has to be accountability to the rules, which means someone must enforce the rules.
There have been times when wars were acts of personal vengeance between rulers of countries, but for the most part, wars are because some who are in power are evil and trying to impose their desires on others. Countries are given the right to defend themselves from those who are evil. War isn't good because it is generally started by those who are not good. But because we are dealing with men, war becomes a necessity.