Changing another person's mind is literally one of the hardest things to do in the world. Think of how many conversations you've ever had in which one of the participants decided the other was right and abandoned their previous views altogether. It almost never happens.
Why? Because even though ideas flit in and out of our heads like mosquitoes, ideas that are believed cling with electromagnetic power. Once we believe an idea we develop an emotional connection to it, not to mention a commitment to it—as if to a person—and often become attached to it with a strength we often don't realize has little to do with the merit of the belief itself. And once we're attached to anything—whether a person, place, thing, or idea—giving it up is extremely hard. We will always grieve over a loss, no matter how small.
CLIMBING A MOUNTAIN IN A SNOWSTORM
Despite the difficulty of changing another person's mind, sometimes it really is necessary to try. Sometimes an idea to which we find a person attached poses a genuine risk to others (believing, for example, Jews, blacks, Muslims, non-Muslims, Americans, or any other collection of individuals who share similar traits are in some way inferior to everyone else, abstracting them into something less than human and giving everyone else the right to treat them with anything other than loving compassion and kindness). Even when the stakes are less dramatic, the effort required is often worth it.
So how can we succeed in changing another person's mind when so much psychology is in play to thwart our efforts? The following strategies may be a reasonable place to start (though are obviously by no means guaranteed):
WHAT IF WE'RE THE ONES WHOSE MIND NEEDS TO CHANGE?
Of course, the foregoing has all been predicated on the assumption that in any given disagreement your idea is the correct one. But if you always and only focus on changing the minds of others and never on your own, you'll find yourself more often than not talking at others rather than with them (and they with you, each of you pausing while the other speaks not so much to listen but to think about what they want to say next), championing ideas rather than exchanging them, and learning nothing. So before you go using any of these techniques on someone else, use them first on yourself.
Try to listen to others as if you hadn't yet decided what you think about the subject under discussion, even if you have. Remove, at least temporarily, your attachment to your own ideas. Give the ideas of others a fair, unbiased hearing, as much as you're able. Notice which arguments cause you to react most negatively and rather than actually reacting negatively ask yourself what's causing your reaction (no easy feat, certainly). What experiences may you have reasoned from poorly to arrive at your beliefs?
If you try as hard as you can to embrace someone else's belief but simply can't become attached to it because you find it inferior by every measure on which you rely, and further your interest in changing their thinking stems from compassion (either to relieve them of suffering or prevent others from suffering at their hands) and not the common desire we all share to have everyone else in the world think exactly as we do—then approach the task of changing their thinking humbly, never allowing yourself to denigrate anyone because they won't listen to your reason. And always remember—especially when you fail—though many beliefs are genuinely dangerous, there's far more room for differences of opinion than we commonly believe.