What an interesting question.
first, accepting that all feelings are natural and normal, but that we have a choice how we deal with them, and how we express them.
for example, when really angry, I can hit you, shout at you, or, tell you that I am angry because of xx, and talk through the problem. They are increasing levels of maturity, showing increasing levels of control. Obviously many of us d a mixture of the 2nd and 3rd, but sometimes the mature thing is to chose to leave a situation, go away and calm down, and then come and talk about it.
second, understanding that our actions effect others, and that we must take responsibility for that reaction. In other words, if I tease and tease and tease at some point the other person is going to get really angry. At which point you had better hope that they are mature as in point 1!!
I want to be clear that this isn't the same as in a victim blaming situation. It is understanding cause and effect.
third having empathy. So, realising that someone else is tired/upset/sad and understanding that feeling, and also (and this is much harder) understanding that at that moment you should do what THEY need, even if that means you should leave them alone.
Part of this is really understanding that you are not the centre of the universe, that there are 2 sides to every story, that most situations are not black and white etc etc.
I actually think many many people are NOT emotionally mature, but we still have to deal with them!