When you're actively engaged in a conflict, what you want is for the other person to really understand your position on the subject at hand. It's easy to forget that the other side wants the same thing -- for you to listen and really hear what they he or she has to say.
Of course, if you dislike another person's personality, it's very easy to develop an us-versus-them mentality. With that kind of approach, conflict becomes a routine problem, and it makes it unlikely that you'll really listen to any valid grievances or ideas that person might have.
So, when the other person is trying to explain his or her perspective on a subject, force yourself to pay attention, and reaffirm that effort by paraphrasing what the person says. Carefully and thoughtfully rephrase their ideas aloud so that they know you hear what they say.
Not only will the other person appreciate your attention, but by repeating their words you may very well gain a better understanding of his or her position. Ultimately, your conversation will be more useful, and in the end, the two of you may develop mutual respect that pays huge dividends in future interaction.