How can you truly resolve family conflicts? Let's start with your own mindset when it comes to conflict in your family. The first step is realizing that whenever we are talking about people, there is going to be conflict. And the more time you spend around someone, the more potential there is for conflict. The Bible says the reason for this is sin, you are born with a nature that wants to put yourself first. I want what I want, when I want it, and how I want it. Having this mindset will help keep you from turning every conflict into a catastrophy.
The second mindset simply involves wanting to see conflict resolved. Some people seem to thrive on conflict, stirring the pot, always having to win the argument or getting the last word. Along with this can be a reluctance to admit fault or to say "I am sorry." Having a mindset that desires peace and harmony in your family relationships, and being willing to do what it takes to resolve it when it happens will go a long way toward see this goal achieved. Remember, you can't change anyone, but you can change how you respond to conflict.
Let me suggest three practical things you can do to resolve conflicts. The first is to simply forgive the person and move on without making it an issue. Overlooking conflict is different than avoiding conflict. When you avoid conflict, you are pretending that there really isn't a problem, you refuse to deal with it. Overlooking conflict and forgiving involves acknowledging to yourself that the other person did do something to offend or hurt you. But you determine that it is not something serious enough to confront them. So you make a concious decision to overlook it, forgive them, and forget it. You don't dwell on it, stew about it, or let it put distance between you and the other person.
Let's say the offense is not something you can overlook, then you need to go to that person and talk about what took place. I suggest that you avoid using statements that begin with the word "you." When you start by taking about what the other person did, you put them on the defensive right away. Instead, use "I and me" statements and focus on how you were hurt or impacted by what happened. If you know that you did something to contribute to the problem, start by confessing that to the other person and ask for their forgiveness. This will help open the door for the them to also respond with humility and admit their part in the conflict. Let's say you honestly can't think of anything you did to contribute to the conflict, you can still say early in the discussion, "If I did anything to offend you, or contribute to this problem, please tell me so I can seek your forgiveness and help resolve this."
Another way to start the conversation is to talk about how much the person means to you. Let them know that your relationship with them is important to you, you value them, and you want to maintain a healthy family. This will let them know that the conversation really isn't as about "you" or "me", but us. How can "we" move past this and maintain a good relationship.
After you "break the ice" by admitting your part or talking about how important your relationship is to you, talk the specifics of how you viewed what happened and its impact on you. Give them the opportunity to voice their view of things, and seek to find common ground. What are the things you agree on, and verbalize them first. Then talk about where you disagree and seek to find a compromise where possible, and extend forgiveness.
Let's say you try to discuss this, and you still can't get it resolved. A third way to resolve conflict is to contact your pastor, a counselor, or some other neutral third party who has experience working with people to serve as a mediator. This person will hear both sides and offer suggestions and ideas on how to work out the problem. Do this before things build up over time and cause bitterness and you and the other person digging in for a long battle of attrition and trench warfare. This will bring nothing but further pain, heartache, and a greater division in your family.
For more information on how to resolve conflict, I recommend you consult Peace Maker ministries, and the books on Peace Making by Ken Sande. There you will find a wealth of information.