Many graduate students have a source of financial support that pays their tuition and a small living stipend. Sources of funding include fellowships (from donors, universities, foundations, government agencies, and industry), employer support, Graduate Assistantships - Research (i.e., money from a faculty member's research grant) and Graduate Assistantships - Teaching/Non-Teaching.
Start looking for funds early. Deadlines for applications vary and can occur before you start your program of study. After you apply, it may take as much as six months or so to review the applications and several more months to actually start receiving the funds.
Ask faculty members, department administrators, and fellow graduate students about available funding. Make an appointment or send an email to the Graduate Award Advisor; look through the listings in The Annual Register of Grant Support, The Grant Register and Foundation Grants to Individuals.
For a research scholarship or fellowship, you will have to write a proposal. You may need to tailor your proposal to the interests and needs of the particular funding agency or program you're applying to, but stick to something you know about and are sincerely interested in.
Write for a general audience, since the people reviewing your application may not be in the same field. Emphasize your goals and why the project you propose to work on is important. Talk as much as you can about how you're going to solve the problem or address the gap in knowledge, and be sure that your proposed methods will satisfy the goals you've set forth. Follow the rules for format, page layout and length, or your application may not even be reviewed.