A grad student should be elected/appointed/coerced each year to be the representative both of the department to the Graduate Student Council and the formal representative of the grad students to the faculty within the department. In the event that no person is willing to perform both of these functions simultaneously, the job can be split up to have one person be the Grad Student Council representative and one person be the department-only rep. Grad Student Council meetings are once a month. The current GSC president should be contacted when the rep changes. His or her name and email address should be on the GSC website at http://www.student.virginia.edu/gsasc/.
Within the department, the grad rep should be a contact with faculty for non-TA related stuff, for example: buying furniture or anything else grads need, mediating disputes, etc. Students are of course free to take personal concerns directly to the faculty themselves, but the grad rep is available as a liaison if desired. For issues which concern the grads as a complete group, it is preferable to have one person represent everyone. It's probably best for this person to be someone in their third year or beyond -- someone who has been around long enough to know how most things work in the department and with whom the faculty are already familiar.
In general, the grad rep should try to keep things running smoothly and make sure that his/her fellow students are aware of their responsibilities. Among other things, the grad rep should do the following: