It frees you to concentrate on your academic work. What you are expected to do is to make good progress toward your degree. Departments choose which students will be offered these awards. There is tremendous variety in the amount they pay. A few will pay all tuition and fees together with a generous stipend.
Fellowships usually are for a specified length of time, one to three years.
Some can be renewed, but most cannot. The conditions should be explained clearly when the award is made. After a fellowship expires, the student is offered an assistantship for which teaching or other work is required. An assistantship usually means that you teach, or assist in research or administrative work. Assistantships sometimes pay enough to live on.