The Academic Proposal
Some application requirements include a request for an academic proposal when research is the primary focus of the project. A typical proposal might include a description of your course of study or project, and why you have chosen this particular institution, country, or setting. You should provide evidence that you are qualified to undertake the program you propose, and that it is consistent with your long-range plans.
For study abroad project proposals, if possible provide evidence of cooperation of the host institution or individuals with whom you propose to work.
For graduate schools admissions essays, discern what is being asked for in each essay and keep the focus on that. Speak mainly about your research interests if that is what they are asking you to address.
For a plan of research essay, devote considerable effort to your "methods" as you need to demonstrate you have a plan, not just a good idea.