Jesus addresses "turning the other cheek" in two sermons -- the "Sermon on the Mount" (recorded in Matthew's gospel) and the "Sermon on the Plain" (recorded in Luke's gospel).
Matthew 5:39 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.
Luke 6:29 And unto him that smiteth thee on the one cheek offer also the other; and him that taketh away thy cloke forbid not to take thy coat also.
What does Jesus mean when He commands us to "turn the other cheek"? What does He not mean?
First, He is not speaking universally of all forms of evil at all times. The Lord's churches are in the business of resisting spiritual evil. "For we wrestle not against flesh and blood, but [we do wrestle] against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places." (Eph. 6:12, Cf. also Jas. 4:7; I Pet. 5:9; II Cor. 10:3-4) Further, He is not abolishing governmental resistance of evil and punishment of evildoers. The Bible teaches this, for example, in Romans 13:4 -- "For he is the minister of God to thee for good. But if thou do that which is evil, be afraid; for he beareth not the sword in vain: for he is the minister of God, a revenger to execute wrath upon him that doeth evil." Finally, Jesus is probably not excluding all instances of self-defense (Cf. e.g. Luke 22:36). "Resist not evil" refers to the kind of evil in the context.
"Turning the other cheek" falls within the context of Jesus addressing misconceptions and misapplications of the law by the Pharisees. Jesus is not come to destroy the law, but is fulfilling it (Matt. 5:17). One must have an understanding that exceeds the hypocritical righteousness of the scribes and Pharisees, which "righteousness" is legalistic and misinterpreted (5:20). Jesus's listeners had heard the Pharisees quote the law and teach on it, but He contrasts what they said with "I say unto you."
Jesus sets forth what the listeners have heard: "An eye for an eye, and a tooth for a tooth." Such a judicial policy in the law was clear and equal. But personal applications of retaliation lay outside the law. Revenge is not legal justice. In fact, says Jesus, "resist not evil," and gives three personal examples of it -- an insult to one's person, an injury to one's estate, and an injustice to one's autonomy. Verse 41 relates "whosoever shall compel thee to go a mile, go with him twain." Matthew 27:32 shows one situation when a person might be compelled -- Roman soldiers compelled Simon of Cyrene to carry Jesus's cross. One compelled to go a mile can freely go the second mile. The man sued at the law may receive the injury to his estate, and then give more rather than counter-sue (v. 40).
In the Sermon on the Mount, remarkably, Jesus says, "whosoever shall smite thee on thy right cheek." Too much or too little may be made of this. One interpretation that makes too much of it can be seen in this example: "In Jesus time and place in history, the left hand was used for 'unclean' purposes. You wouldn't use your left hand to purchase food, shake someone’s hand, OR even strike someone. It would be a shameful act to use your left hand for these things." Followed through (see link), such an interpretation posits that if someone hit you on the right cheek you offer him the left, forcing him to acknowledge you as equal (that is, by slapping you with his right/clean hand). This goes too far and clouds the simple lesson. If Jesus so intended, why did He not mention the right hand in the Sermon on the Plain as well? But the mention of the right hand is a clue that shows us that an insult is intended -- as opposed to physical assault. In a predominantly right-handed world, a back-handed slap would normally be required for a right-handed person to strike another person on the right cheek. Clearly, though, the spirit of the command does not depend on which cheek is smitten.
Retaliation only escalates and multiplies the amount of evil. In following Christ, we choose not to retaliate. Revenge is deferred to the realm and oversight of God (Romans 12:19), while the Christian follows the example of the Lord Jesus Christ "Who, when he was reviled, reviled not again; when he suffered, he threatened not; but committed himself to him that judgeth righteously" (I Peter 2:23). Such follow-ship is direct evidence of kinship to the heavenly Father (Matt. 5:45).
God settles all accounts.