Determine what your topic is going to be. In order to begin researching your topic, you need a preliminary starting point. Choose a topic that has some controversial angles. You do not want to choose a topic that has no conflicting bodies of work surrounding it, as a good argumentative paper takes a position on an existing controversy.
Research your topic. Using Google Scholar or other search engines that index peer reviewed articles, try to discover the research angles or schools of thought regarding your topic. Search your school library's catalog as well, and use databases that are linked through your school library's website. for the most comprehensive search results. Typically for any controversial topic there will be two or three schools of thought with research to back them up.
Choose the school of thought that makes the most sense to you, and that you find has the most evidence backing it up. This is going to be the research you will be aligning yourself with, so get a handle on its tenets. You should read at least five outside sources from peer reviewed journals within your school of thought, and be prepared to cite three. Check with your instructor for specific requirements regarding how many sources you need to use.
Write an introduction. Your introduction should contain your thesis, and a some brief informative statements to provide context about the subject of your thesis. Your introduction's length will vary, depending on the length of the paper, but it should be at least three sentences. Conclude your introduction with your thesis statement. Remember, a thesis is a falsifiable statement; that is, your thesis is not just a summary, but needs to present your clear and unequivocal position on your topic and its truth or untruth is contingent upon evidence.
Begin writing your reasoning section. Here you will convey the reasoning for your thesis statement. This is the "argument" portion of your paper, so you should spend the most time here, writing three to five paragraphs. Make it clear to the reader why you have chosen the thesis you have. Your thesis should serve as something of an unspoken conclusion to your reasoning section.
Cite at least one popular counterarguments to the reasons you have given for forming your thesis. It may be tempting to convey these counterarguments as being weaker than they really are; avoid doing this. Stay as true to the authors of the counterarguments as you can, and clearly state what they find incorrect about your thesis or the school of thought that espouses your thesis.
Write a response section. This is a direct response, in your own words, to the counterarguments you have cited in the previous section. Give credit to the authors; don't dismiss their arguments out-of-hand. Give their counterarguments the weight they are due, but convey why you have not chosen to side with their school of thought.
Finish your paper by writing a conclusion. You should summarize your thesis and highlight a few of your main points. Don't introduce any new ideas, but you can make suggestions for future research and indicate any limitations you may have in your argument. Clearly conclude your paper, wrapping up any loose ends and leaving the reader with lasting image that drive home the message of your argument. Your conclusion should be no more than a paragraph long.