Ok. I think @VolvoGirl is slightly incorrect, in that, while an organization may be exempt from social security tax, its employees rarely are. There are exceptions; for example, in NY, police, corrections workers and other peace officers are covered by a state pension instead of social security. If this applied to you, your employer would be able to provide all the details needed.
However, students are entirely different. Students are exempt from social security if they are employed by their university in connection with their studies. For example, a student employed as a TA in their degree field would be exempt from Social Security on their TA wages; a student employed in the cafeteria would generally not be exempt from social security. Graduate students who are paid a stipend (usually in the various sciences) are also exempt from social security. You don't need any extra paperwork or to pay any extra tax. Just enter your W-2 and go on to the next topic.
Now, the (slight) downside to this is that you don't get social security credits toward retirement or disability. I spent 6 years as a graduate student and 6 years as a post-doc and was exempt from social security for that time. When I retire, my benefits are based on my 35 highest-earning years over my lifetime. Because I spent so much time exempt, some of those zero years end up in my 35 year look-back and drag down my average earnings and therefore my social security benefit -- of course that's assuming that social security is even solvent when I retire. If your university allows you to contribute to some other kind of retirement plan, such as a 403(b), you should take the opportunity to invest at least something for your future.
