• When starting a new job, you'll need to give your SSN to your employer so that they can report your income to the IRS and wages to the Social Security Administration.
• When opening an account with any U. S. financial institution.
• When applying for a federal loan, such as a federal student loan.
• When applying for certain types of public assistance, such as unemployment benefits.
• When enrolling in Medicare (you apply for Medicare through the Social Security Administration).
• When applying for a passport.
• On your tax return, so that the IRS can match the income you report to that reported by your employer(s) and financial institutions.
• To get a driver's license.

As for when NOT to give out your SSN, as well as more complete explanations of the above, check out this article.