First of all, family problems can only be "worked through." They cannot be "solved." The issue probably more about learning to recognize and appreciate what the other person thinks and feels about the contentious issue.
This post is focused on adult family situations. Using the following tactics with a 2-year-old are less effective, though quite comical as my wife and I have learned.
- Listen more than you talk. You may have heard the adage, "you have two ears and one mouth for a reason." This is crucial in interpersonal communication, and especially true in conflict resolution. It will both allow you to genuinely hear what the other person is saying, (provided you're actually listening instead of formulating a really biting comeback) and it will also help the other person to feel that they've been heard.
- Affirm the other's thought and feelings. Remember that you don't have to agree with them, and probably don't, about why they're upset. The fact is, they are upset for some reason, they attribute that, at least in part, to you, and whether you agree or not, they do feel that way. Even if you're completely innocent (which none of us ever are) you can at least apologize for the person having gotten that impression from you, or for the fact that they feel bad at all. An apology goes a long way to making amends. It also gives power to the other party, and this is key. This is a conflict RESOLUTION, not a who-is-wronger competition. In a healthy relationship, both people give relational power to the other. This is called vulnerability and it is important. Apologies cool tempers and act as a verbal olive branch to the other person.
- Look at yourself from their perspective. Imagine if you had done to them what they did to you. (But you would never do such a thing because you're not that...) Just imagine if the tables were turned. Now imagine how you would want that person to respond to you, NOT how you think they'd be justified in responding to you. My brother used to joke that the "golden rule" was "do unto others as they do unto you," instead of the more agonizingly painful and frustrating, "treat others as you wish they'd treat you." I have some very close friends who I didn't invite to my wedding. It was not an intentional slight, but the wedding was far away and we had to cut down the guest list and I just somehow thought they wouldn't really care anyway. When I spoke to them next and told them I got married, they were obviously, and rightfully, hurt that they were not invited. However, because my friends are people of high character, they didn't spend time making me feel more guilty. Instead, they treated me with grace.
These three steps are far from exhaustive, but they'll make a great starting point. Remember that working through issues is a long and arduous process. Depending on the issue, it may take months or even years to fully move beyond it. But by practicing these three simple (but really hard sometimes) behaviors, you'll find that over time conflict resolution will become easier and more productive.
You'll also find that when you do these things, others will likely follow suit. Don't expect it right away, but commonly an apology elicits an apology, and when a problem is heard, it gets a little smaller. Coming from a guy who's screwed up a lot and forgiven a lot, it definitely gets easier over time.