If you find yourself irrationally angry at something a coworker is doing, that is not the time to confront her. Wait until you've calmed down and have a clear head. You may decide that you don't even want to confront the person. You don't want to bring up a problem in the presence of twenty coworkers in the break room. However, if you think the confrontation could go badly, you might want to bring someone else along to help mediate or to at least witness the confrontation. You need to be polite but assertive. That is, you don't want to downplay the problem, but you don't want your coworker to feel like you are attacking him. For instance, if the problem is a coworker is playing music too loud, try approaching it this way: "Can I speak to you for a minute? I'm sorry to bother you about this problem, but your music being so loud really distracts me from my work. Is it possible we can find a solution to this problem?" Don't drag in everything you think is wrong with the person. Stick to the facts. Also, don't name call or degrade the other person, as those actions are not professional and reflect badly on you. You can also lower the tension of the situation by cracking a joke. Usually, it's best to make the joke at your own expense, sort of putting you and the other person on the same playing field. That is, when you're bring up something negative about another person, you are creating a negative tension between you two. By cracking a joke, you can help erase some of that tension. For instance, if you've noticed they left a mess again in the kitchen, you could say something such as "I noticed you left a mess in the kitchen. Would you mind cleaning it up? Don't worry, I'm just as bad. I'm such a slob sometimes; they may need to send in an excavation team for my dirty dishes at home, haha." When you need to bring up something negative, try starting and ending positive. That way, your coworker doesn't go on the defensive when you first start talking to him, and he won't leave the conversation with a sour taste in his mouth. For instance, if your coworker won't be quiet about politics, maybe you could say, "I love how enthusiastic you are about your political beliefs. However, you are making some people uncomfortable who don't share your views. Could you maybe save that conversation for after work? It's great how passionate you are; few people care that much." In every relationship, you have to give as much as you take. Therefore, if you're demanding that your coworker give up something, try to offer something in return. For instance, if you ask your coworker to turn her music down, try saying you'll also wear headphones to help the problem. Trying to out-annoy the other person is likely to end badly. For instance, if you don't like that someone is playing her music too loudly, increasing the volume on your music isn't going to solve the problem. Plus, it will get other coworkers annoyed at you. If direct confrontation isn't your thing, you can still find away to approach the problem. One good way to do that is to bring it up in a work meeting as a general office problem. For instance, you could say, "I've noticed the noise level in the office has gone up dramatically lately. Could everyone please pay attention to the amount of noise you're creating?"

Summary:
Wait until you're calm. Don't approach the person publicly. Approach your coworker with the problem. Keep it professional. Try a little humor. Bookend the conversation with positive notes. Don't just demand. Don't retaliate. Take a sideways approach.