A social security number (SSN) is an identification number that lets the US government determine what taxes were paid to the government by your employer. F-1 students are only eligible for a SSN when authorized for work (on campus, CPT, or OPT). J-1 students can apply without work authorization but a SSN is not needed unless you will be working. 

International students will need their employer and International Student Services to complete the Instructions for applying for the SSN.