Your U.S. SSN is linked to your credit history (you should get an annual credit report). Treat it as confidential information and avoid giving it out unnecessarily. If you feel uncomfortable giving out your SSN, ask why it is needed and what happens if you refuse to give your number.
Generally your bank, employer, the Internal Revenue Service (for tax purposes), and the Department of Motor Vehicles (for your license application) are the ones who need your SSN. Utility companies, cell phone providers, prospective landlords, and credit card companies (for your credit card applications) will ask for your SSN. Be wary of any email requests for personal information (bank account number, SSN, etc.), which appear to be sent from these agencies. Please be aware that U.S. government agencies, banks, the University, and any other legitimate entities would never request personal information over email. If you respond to such emails, you may become a victim of identity theft.