I can't speak to program requirements, but I can say something about the kind of students the programs seek—grad programs look for people who show evidence they can DO as well as learn, At the academic end that means, grad programs seek people who can: 1) communicate clearly through both writing and speech; 2) organize both their own time and activities as well as information; 3) contribute to the collective work of the subject by transforming ideas, data sources, methods of analysis. In non-academic positions that might mean transferring the ideas and methods of the field to problems in administration, organization, leadership, non-academic research, policy-making, etc., etc., etc.
Where undergraduate education is principally based on taking instruction and acquiring information, graduate programs seek self-starters, people who can generate their own ideas, figure out where they can find the data to test those ideas and employ the methods of the subject as well as of allied fields and perhaps their own innovations to analyze the data that tests the ideas. Being well-read in ones subject, abreast of the latest publications and connected to peers in the field are all desired qualities. (It’s been my experience that in graduate school you will learn more from your peers than from your professors.)
The honors thesis or senior essay that may have been the culmination of your undergraduate education is the starting point for a graduate one. If you have, or have had, the chance to conduct such a project, emphasize it in your graduate school applications. Point out your skills in problem selection, data collection, analysis, writing and basic self-organizing. Let your referees know those are the points you would like highlighted in their recommendations—they will probably know this, but some might not say it unless prompted. If you have conducted non-academic research and analysis—maybe as an intern you were asked to collect data for your supervisor to use in policy-making or task-setting—highlight this too. This last points to the kinds of internships to seek. A solid demonstration of motivation and skills will take you further than then all the best grades and test scores.
That said, programs too often have inalterable bureaucratic requirements: know what they are and be prepared to meet them. I had held my doctorate for ten years when I applied for entry into a master’s degree program. While completing a PhD was surely the better evidence of my graduate school abilities, I was nonetheless required to submit twenty-plus year old undergraduate transcripts.