Question: "What did Jesus mean when He instructed us to turn the other cheek?"
Answer: The entire section of Jesusí Sermon on the Mount in which this verse is found can be understood as one where Jesus actually serves to elevate the importance of following Israel's moral law beyond the letter to the spirit of the law. Much of the material therein complements the nature of His coming characterized by mercy, sacrificial love, and longsuffering toward sinners while at the same time affirming the "last is first" principle upon which the kingdom of God is based. For instance, we are told to go the extra mile for someone who abuses us and to pray for enemies instead of resisting them. All of this can be generally summarized by saying we need to be pure inside and out and should be as accommodating as possible for the sake of a lost world.
To "turn the other cheek," does not imply pacifism, nor does it mean we place ourselves or others in mortal danger. Like the principle of the eye for an eye and tooth for a tooth in Matthew 5:38, turning the other cheek refers to personal retaliation, not criminal offenses or acts of military aggression. Clearly, Jesus did not mean to negate all Godís laws and injunctions protecting us against violent crime or invading armies. Rather, Jesus is speaking here of the principle of non-retaliation to affronts against our own dignity, as well as lawsuits to gain oneís personal assets (v. 40), infringements on oneís liberty (v. 41), and violations of property rights (v. 42). He was calling for a full surrender of all personal rights.
Turning the other cheek means not to return insult for insult in retaliation, which is what most people expect and how worldly people act. Responding to hatred with love just might grab someone's attention and afford us a chance to share the gospel. When we respond in a manner that is unnatural, it displays the supernatural power of the indwelling Holy Spirit. Jesus was the perfect example because He was silent before His accusers and did not call down revenge from heaven on those who crucified Him.