When I applied to graduate school, I was really surprised by the differences in the ways universities handle funding and appointments for graduate students. In many cases, applicants have to be more than a little savvy and be willing to ask potentially uncomfortable questions about money. Some universities (or schools within a university) require prospective graduate students to fill out applications for assistantships, that is, an application separate from the application to admission to the program or school. Many programs, however, will automatically consider you for a research or teaching assistantship based on your program application materials. Some programs/schools have competitive fellowships for applicants, so be sure to call the programs to which you apply and ask explicitly.
No matter what avenue you pursue for funding your graduate program, don’t be afraid to ask questions. I think the most appropriate times to ask is both before you apply to the program/school (scholarships/fellowships – some have early deadlines) and then once you have received a letter of admission into the program (assistantships). If the program or school doesn’t send information with your acceptance letter, pick up the phone or send an email to the contact for your program and ask them about assistantships and fellowships.
Most assistantships come in two forms: research assistantships (RA) or teaching assistantships (TA). For RAs, you will probably be expected to help a faculty member with some aspect of a research project(s): doing literature searches, conducting lab or field experiments, collecting and/or analyzing data, and in some cases, writing. For TAs, you will teach one or more sections of an undergraduate course in your program or school. If you don’t have previous teaching experience, this can be pretty intimidating. Most universities now have supports in place to help TAs get oriented to teaching, but honestly, your best resources for teaching are faculty members and the seasoned graduate students in your program. No matter what type of assistantship you receive, you can expect a tuition waiver (partial or full) and a (very) modest stipend for teaching. Most graduate students find it difficult to live on the stipend they receive, but to me, the tuition waiver is what makes the assistantship fiscally valuable. Without it, many graduate students would be strapped with monumental student loans, especially if you are coming to the program as an out-of-state resident. So you will find that you probably will have to take out some student loans, but having the tuition wavier makes the amount of loans more reasonable. Finally, from a professional standpoint, assistantships are valuable because they provide you with experience with teaching and/or research that becomes invaluable as you enter the job market.