The word "assistantship" suits this type of award perfectly: In exchange for your assistance, you'll be assisted in paying your tuition! Plus, you'll get the chance to get involved in the internal workings of your academic department. This is the most common form of graduate aid.
Teaching assistantships
In exchange for teaching within your field, you are given a salary and/or tuition help. The specifics will depend on the school, but duties may include grading papers, monitoring labs, or teaching courses. Large public universities use many TAs.
Research assistantships
These are similar to teaching assistantships, but involve research. (If you figured that out, you're one step ahead!) In most cases, you receive funding in exchange for doing work in your discipline. This type of award is common in the sciences and social sciences.
Remember way back in college, there were advisers who kept the dorms in order? In most cases, these folks were grad students who supervised in exchange for tuition help. This option is primarily given to students studying psychology, counseling, and social work.
Fellowships are basically scholarships for grad school. You aren't obligated to do anything in return - except go to school. You usually have to show exceptional talent in your field to qualify, though there are grants based on need. In addition to institutional awards, there are also grants and fellowships awarded by outside sources.