You really can't expect me to do anything else by now, but this in this exact order:
1. Write a long letter (after giving yourself and that person some time to cool off and understand perspectives and mistakes). And remember, never too big or small or right or wrong to apologise, because you did play a part in the fight.
2. Make them a brilliant meal – usually two course with a nice warm beverage.
Any conflict we have happens happens or gets exaggerated when we fail to remember that both of us are facing the problem instead of facing each other. When things get heated up, I tend to stop talking and ask for some time, calm my mind and think about what just went down. Then I call the other person up or meet them and listen to them.
The final show is that we need to control our anger and make good decisions.
Communication, communication, communication. You have to be honest and transparent about what bothered you, why it bothered you and how to avoid it in the future. If you don't talk things out and just sweep them under the rug you'll never solve anything and the problems will continue to re-occur.
Giving and taking the cooling time firstly. After this you are in a position to see what went wrong and how can you make things better. If you operate with this thought of working out things, by conversation and understanding the other person you can resolve any conflict. Then of course saying some kind words or giving a treat is an add on.