Choose a theme for your essay and formulate a thesis statement. Your teacher may have already given you a topic to write about, either as a list or limiting you to the mythology currently being studied in your lesson plan. If you have a broader range, look for myths that hold your interest so that the paper will feel less like a chore to write.
Research the mythology within your topic thoroughly. Your research options include going to the library, researching online or contacting mythology and folklore departments at your local university for help. Collect as many references as you can so that your paper has a solid foundation. In most cases, teachers assign a minimum amount of sources to use, so if nothing else, collect the minimum amount.
Take notes on your chosen topic as you research from your references. These notes will help you decide more clearly on the direction you want your paper to go. For instance, you may find a fact about your myth that you were unaware of which could change your original idea around some.
Outline your paper. Outlines involve making a preliminary sketch of your essay so that you can be sure that you fulfill all requirements. Your outline should consist of brief sentences to describe each paragraph of your paper. You can rearrange your ideas so that the paper reads more cohesively before beginning to actually write the paper.
Write the body of your essay first. Often times, people get stuck when writing essays and research papers because they begin with the introduction. Introductions and conclusions can be saved for last because it will be easier to write a summary of your essay's main points after you write them.
Incorporate other myths into your paper. Often times, you will be able to find myths that have the same theme as each other. By putting them into your paper, you show a clear understanding of the myth in question.
Add in historical references for the mythology. Knowing the historical background of the time period in which the myth was created can lend to understanding the mythology better. Try discussing any historical points that you feel contributed to the creation of the myth.
Edit your paper, or have someone else edit it for you. For best results, choose someone to edit that knows about the myth you're writing about so that they can tell you whether your points make sense. Alternately, selecting someone unfamiliar with your myth helps to see if your points are clear enough for anyone to understand them.
Write a final draft of your essay to be turned in. The final draft should include any editing changes made earlier so that the paper reads well.