The Graduate Student Association is the official governing body of graduate and professional students at UCSD. The association advocates for students' rights at all levels through communication and cooperation with all branches of both the student body and university administration (local, state, and federal legislators as well!). In addition, the GSA sponsors many social events designed just for graduate students. Keep an eye on the mathgrad email list for updates about these events from your trusty GSA representatives.
The GSA is organized into two main parts: the Council and the Executive Committee. The Council is comprised of representatives from each graduate program (how many representatives a department gets depends on its size; the math department currently has 3 reps). The Executive Committee consists of elected council members who take on the senior leadership positions and oversee all aspects of the GSA. At a typical meeting, the Executive Committee leads the Council through a series of discussions and votes regarding wide varieties of issues affecting graduate students campuswide.
One of the most important ways that the GSA helps individual departments is by funding department events. A certain amount of money is set aside for each department just to spend on its own grad students! For instance, the Food For Thought talks are fully funded by the GSA.
Any graduate student is eligible to be a representative, so if you are interested, just check to see who the current reps are and talk to them about it. A list of current reps is always maintained at http://gsa.ucsd.edu. There you can also read as much as you care to about the constitution, by-laws, and inner workings of the GSA as a governing body.