When the dust from the fight has settled and you are more motivated to resolve the issue rather than rip each other to shreds, follow these simple rules for successful conflict resolution:
Set the stage.
It is important to decide on the specific time and place to meet in order to discuss the issue. Even if you decide to talk right now it is important to set the stage by asking your partner if they are ready and willing to talk now. If they are not ready now, then you can decide on another date that is convenient for both of you.
Decide on taking turns.
Once you are both in a place where you will not be disturbed (it is better to turn off your phones) it helps to agree on taking turns. This means that while one person talks, another one listens without interrupting. It is extremely important not to interrupt your partner when they talk even if you think that what they are saying is invalid. You will have a chance to point that out when it is your turn to speak.
"I" talk instead of "You" talk.
When it is your turn to speak, rather than saying "You are so and so" say something like "I feel hurt/bad/uncomfortable when you do this and that". This style takes away the blame but still gets the message across. It has been repeatedly shown that people are much more open to criticism presented this way.
Summarize main points.
It is very common that people start thinking of comebacks to what their partner is saying rather than paying attention. When it is your turn to listen, do listen! I mean it, do not think about what you are going to say next! Just try to really understand their point. When he/she is finished summarize their main points. This gives your partner a clear indication that you have in fact listened. At the end you can also ask "Do I understand your points correctly?" This gives your partner a chance to add something in case you missed it or elaborate more on what they said. More often than not your partner will be grateful and will return the favor by listening to you.
You will be pleasantly surprised that by following these relatively simple steps most of your fights will dissolve themselves. If not, then the fifth step is necessary.
Brainstorm for possible solutions
Once you are both clear on the issues that you and your partner have presented you can both think of possible solutions to avoid similar confrontations in the future. Usually, after 3 or 4 possible scenarios are considered, it is possible to find the one that is most suitable for your situation.
Remember, you don't have to agree on all issues. It's okay to have disagreements in relationships. That is why many couples "agree to disagree" on some issues. There is no need to create a relationship that is free of conflict.