Tom explained this nicely. To add one more point, you won't receive a tuition bill, your advisor will. Doing a PhD is less like school as you've experienced so far, and more like working for a small company run by your advisor. The advisor pays your tuition to the university from his or her grant funding, along with what the university charges as overhead. It's almost as if each research group is a franchise, and the university charges a fee so that researchers can use their brand name.
As others have pointed out, schools will describe their stipends, which are designed to cover your living expenses and do so comfortably but simply. They also provide some sort of health care benefits. Serving as a TA is usually required. And, NIH or NSF fellowships can increase both your stipend and provide funding for your research. These fellowships are competitive, so obtaining one is also good for your C.V.