"To the married I give this command (not I, but the Lord): A wife must not separate from her husband. But if she does, she must remain unmarried or else be reconciled to her husband. And a husband must not divorce his wife." NIV translation
Paul reminds the Corinthians that God does not like divorce and so the married should stay married.
"To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her. And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him. For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy." NIV translation
Have you ever heard of staying in a marriage for the children's sake? That is what Paul is saying to do in this situation but we have to understand that this is from him and not a "command of God". This is also specific to a marriage between a believer and a non-Christian.
"But if the unbeliever leaves, let him do so. A believing man or woman is not bound in such circumstances; God has called us to live in peace. How do you know, wife, whether you will save your husband? Or, how do you know, husband, whether you will save your wife?" NIV translation
We see, from this passage, that the home is to be a place of love not a battle zone. Paul tells the believer that, if their non-Christian spouse wants to leave, then let them go but, if they want to stay, show them the love of Jesus in hopes that they might be saved. There are many who will take this passage and try to force Christians to stay in a marriage that does not honor God but we must remember that God has never asked His people to live in a "marriage hell".
"Nevertheless, each one should retain the place in life that the Lord assigned to him and to which God has called him. This is the rule I lay down in all the churches." NIV translation
This may seem odd as Paul tells us the "rule I lay down" but this rule is freedom. Paul is telling us that, when we accept the grace of God then we do not have to do anything else to be saved.