"Ye have heard that it hath been said, An eye for an eye, and a tooth for a tooth. But I say unto you that ye resist not evil: but whosoever shall smite thee on thy right cheek, turn to him the other also." (Matthew 5:38-39a, emphasis mine)
By the way, in case you ever need it, the word for "turning to the right" is: dextrorotatory. You might have to shell out five bucks to use it though.
Jesus must be referring to the Roman law of noxiam sarcire. Latin for "paying for the damages", literally. And as Jerusalem had been annexed by the Roman government, you can understand how He'd preface that passage with "ye have heard". The law went something like this: whenever someone was wronged and suffered damages, either to property or person, the father of the offender was able to make restitution to the offended party by literally giving up his son to them to do with him what they would. When Jesus says "I say unto you that ye resist not evil", it's like Jesus is reiterating Moses' declaration from Exodus, chapter 14 (verse 14): "The Lord shall fight for you, and ye shall hold your peace." One of the many reasons that Jesus was silent before his accusers.
To be sure, there's a fine line between knowing when to stand up for what's rightfully yours—yours either through common decency or legally—and letting people walk on you, take advantage of you and generally treat you as they (mis)treated Jesus. Just know though, that in everything we're confronted with in suffering, God gave up His Son to them and theirs in order to take the punishment in its entirety. The difference is that Jesus never did anything wrong—we were the offenders toward God, and many times, toward other people.
"He that spared not His own Son but delivered Him up for us all..." (Romans 8:32a)
Apparently, one-in-three people the world over identify themselves as Christian. The sheer statistical weight would seem to indicate that the world should be in a whole lot better shape than it is—provided the Christians were doing their job in service to the Lord, whatever that job may be. Factor in the issue of suffering and persecution, and things even out to a degree. See, in some ways, it's not about our success only in appealing to the non-believer. If you see someone who is successful, whether they've encountered setbacks or not, they end in being an inspiration to that end only. But when unbelievers see you suffer and then remain joyful and rejoicing? That's when the Holy Spirit can really get to the hard and impossible (without Him) work of convicting their hearts.
Mohandas (Mahatma) Gandhi resisted British colonialism in India. His famous statement "An eye for an eye leaves the whole world blind." pays a slight homage to Jesus. India gained its independence from England (who annexed it in much the same way as Rome did Jerusalem) in 1949, a year after Gandhi's passing. While he respected and sought in many ways to follow Jesus' teachings, there is much debate over whether or not he was Christian and it's not anyone's place to speculate. In response to a missionary's probing question, Gandhi also said: "Oh, I don't reject your Christ, it's just that so many of you Christians are so unlike your Christ." While wayward and/or arrogant Christians are not enough to warrant an unbeliever's rejection of the God that they supposedly represent, they're not doing Jesus any favors either. This is why Jesus said to submit to suffering—to "turn the other cheek". Lay down any notion of self-justification and self-reciprocation. God says He'll defend you. He proved it by sending His Son to die in our stead.
"Owe no man any thing, but to love one another: for he that loveth another hath fulfilled the law. Love worketh no ill to his neighbour: therefore love is the fulfilling of the law." (Romans 13:8, 10)