Proof of citizenship is necessary when dealing with all levels of U.S. government. The main proof of citizenship is the U.S. Passport.
However, people who wish to apply for a passport will need to submit certain documents that are accepted as evidence of U.S. citizenship. Generally, a birth certificate or naturalization paperwork can be used as proof of citizenship to obtain a passport.
If the applicant is renewing a passport, the old passport will suffice. Proof of citizenship will only be required for first time passport applicants. The birth certificate demonstrates proof of citizenship and is one of the reasons why it is so important.
Also, the Consular Report of Birth Abroad or Certification of Birth (CRBA) is mandatory for persons who were born abroad to U.S. citizens.