How come in one place the Bible says "an eye for an eye" and in another it says we should "turn the other cheek?"
While this may appear to be a contradiction, the difference is when the two teachings were given. One of the secrets of "rightly dividing the word of truth" is to know when it was written. It is significant because the Bible is divided into dispensations. A dispensation is a slice of time when God administered to His people in a certain way.
Each dispensation has its own characteristics and are separated by dramatic events. In each dispensation the people were given certain responsibilities or testings. For example, in the Garden of Eden, during the dispensation of innocence, the test was to not eat of the tree of the knowledge of good and evil (Genesis 2:16-17). When mankind failed the test (a dramatic event), we were driven from the garden and plunged into the dispensation of conscience.
Now, back to your question. The first teaching you mention, "An eye for an eye..." is found in Exodus 21:24. The book of Exodus is written during the dispensation of the law. The "turn the other cheek" portion is found in Jesus' sermon on the mount (Matthew 5:39), embedded in the dispensation of grace, which is the slice of time we are currently in.
The dispensation of the law came to an end with the coming of Jesus Christ about 2000 years ago. We see Him beginning to usher in the age of grace in the account of the woman caught in adultery (John 8:1-11). Some men brought a woman to Jesus, who was caught in the very act of adultery. According to the law (Leviticus 20:10), she was to be put to death. Jesus never argued for her innocence, condoned her sin, or disputed the teaching of the law. He just extended grace to her.
In the dispensation of grace we find ourselves in now, the law serves a very important purpose. It shows us that we are sinners and that we fall short of the standard set before us (Romans 3:20) by a holy and perfect God. Because of the law we realize our need of a savior (Galations 3:24). That Savior is Christ Jesus.
When reading the Bible, it is vital to understand when it was written, and who it was written to, as well as the entire context. This is all a part of good Bible study and being a student of the Word of God. It takes effort and diligence (which is why most lack). Scripture itself says, "Study to shew thyself approved unto God, a workman who needeth not to be ashamed, rightly dividing the Word of truth." (2 Timothy 2:15).
I hope this helps in your future study of God's Word.
God Bless You,
TheAnswerMan