When writing a résumé for a fellowship application, you should consider many of the same factors as when writing a résumé for a job, and tailor it accordingly. The aim is to make it easy for the reader to see a good fit between the education and experiences you've had and what you're applying to do. The Office of Career Strategy offers valuable advice, as well as samples. A fellowship résumé, however, might well look a little different:
If you are asked to follow a particular format, do. For example, if you are asked for a one-page résumé, don’t submit two (or more) pages. Any pages after the first will not be read, which means that the information you’ve put there will not be known to the committee. (Even if more than one page is permissible, the most relevant and important information should ideally be on the first page.) Finally, how closely you follow directions is a key indicator of how seriously you take the application process.
A résumé for a fellowship application might not look quite like a résumé for a job application, since a fellowship committee may be looking for rather different things than a potential employer hopes to see. The selection criteria for a given fellowship and the experience and qualifications that give you relevant preparation for your particular proposal are good things to consider.
If you are applying for funding for study or research, an academic curriculum vitae (cv) will likely be more helpful than the résumé you wrote for a business internship, and vice versa. A cv should highlight, for example, academic awards, research experience, the name of the faculty adviser for your (proposed) senior project, the title of that senior project, etc., whereas a résumé for a business internship may focus on other skills and experience relevant to the position for which you are applying. But remember that experiences you might think of as purely academic might involve skills that are transferable to other fields and industries: think critically about the things you have done and how those experiences might apply to your proposal.
Tip: What does an academic cv look like? Start by looking on your department's website or that of a related academic department for examples posted by faculty and perhaps graduate students (or ask your TF). Don't worry: no one expects an undergraduate's cv to look like that of a senior faculty member, but you want to get a sense of the conventions of your field.
While traditional job résumé headings like “Work Experience” and “Extracurricular Activities” are fine for many fellowship résumés, you might also consider grouping items thematically. You might find that this better demonstrates the background you bring as an applicant for a particular fellowship or a specific proposal, and it might also help you stand out from the crowd. A heading like “Journalism” or “Public Interest Law” could combine your relevant extracurricular and work experiences to show how you are prepared to embark on a particular proposal; and having headings such as “Community Service” or “Leadership” could make it easier to see how you qualify for a particular fellowship.
Consider whether or not items on your résumé will make sense to general readers. Many selection committees will include members who are not experts in your major or field of interest, so it makes sense to aim your application at educated non-specialists. (Some committees, on the other hand, are more specialized, so your résumé could be, too: fellowships like the Goldwater or Churchill or Yale's Tetelman are only for the sciences, technology, engineering, and mathematics, and so those committees will be comprised of scholars in STEM fields; while the Hays-Brandeis is for visual arts, art history, and art conservation, so that committee features artists, arts and art history faculty.)
When applying for outside fellowships, consider whether or not items on your résumé will make sense to readers outside the Yale community. Remember that an external selection committee will not know what a FroCo does, what a Richter Fellowship is for, or what Dwight Hall is. Providing information about these Yale-specific elements will help the committee better understand your qualifications and experience.
Your résumé should be an accurate representation of you, your experience and qualifications, but tailoring it to the application in hand is helpful. You want to make it as easy as possible for the individual reader or committee to see what you bring to the table as an applicant and why you might be a good fit for their fellowship.