As with choosing where to apply to for undergrad, figuring out where you want to apply to grad school is daunting. I was lucky to have a very hands-on undergrad advisor, who helped me organize my thoughts. Together, we came up with a "hitlist" of schools and potential advisors - I modified this hitlist into the spreadsheet below, so I could keep better track of which programs I had looked at, who I had contacted, when the deadlines were, etc. You should try to reach out to potential advisors a few months ahead of the application deadline, to check whether they are (1) interested in your work, (2) taking new students, and (3) able to fund new students. Below the spreadsheet, I've included a template email that I made for the current paleo students at my alma mater; of course, tweak the details (highlighted parts) to represent your own background, and try to keep your message brief. Also, consider reaching out to current students to get an honest assessment of the lab environment, institution, surrounding community, etc.
Do your best to start working on application materials well in advance of the deadlines, and ask an advisor or other trusted mentor to review your essays. Always pay very close attention to formatting guidelines and word/character count limits - you don't want your application to be rejected because of the wrong font size.Â
Generally, grad school applications require one or two essays that tend to be very similar in topic and structure to those required by the NSF GRFP. If you intend to apply for an NSF GRFP as an undergrad, I recommend using those essays as templates for your grad school applications (they usually also ask for "research proposal" and "personal statement" essays) - just butcher them up to fit the exact requirements of each school, and tweak details to be relevant to the institution you're applying to. If you are not applying for an NSF GRFP, consider using the essay(s) written for the institution requiring the most detail as your chopping-block essays - regardless, working from a template, rather than writing a completely new essay each time (which, alas, is sometimes unavoidable) will cut down on time significantly.
Below are examples of typical grad school application essay prompts, one research-focused and one personal. A good way to approach prompts (and organize your essay) is to break them down into the components that they are asking of you. Here, I've numbered the components within each prompt, and highlighted other important details.
Here are my own essays, "Past Research Experience" and "Proposed Research Interests" (both were submitted as a single pdf), that I wrote for my application to AMNH-RGGS. RGGS asked me to include references/citations, but not all applications require that. If they do, make sure to note whether or not they count towards the total page/word count.
It is important to keep in mind that, ultimately, admission decisions are often made for reasons entirely beyond your control. Having an application rejected is far from fun, but it does not necessarily mean that you weren't good enough - sometimes it just means that it was not that advisor's turn to pick a student, that the department did not have funding for students that year... etc. If it happens (and it does happen to everyone), it can be helpful to reach out to the admissions committee and ask for feedback to inform future applications.