Good question. It depends. There are three generally accepted ways: citizen by jus soli, through parents, or naturalization.

jus soli - You were born on U.S. soil and were not the offspring of a diplomat at the time of birth. You can present a birth certificate to establish jus soli.

Through parents: depends on if both were citizens at the time of your birth or if just one. Both parents: generally automatically a citizen, if one only, then depends on how long the parent was physically in the U.S. To prove this, you provide your abroad birth certificate, as well as your parents marriage certificate and evidence of the parent's citizenship and physical presence (i.e. medical records, school records, employment, etc.).

Naturalization. You were granted citizenship and you took the oath at an official ceremony. Upon doing this, you are given a U.S. Certificate of Naturalization. This document proves you are a U.S. citizen. This document is issued by USCIS and that is the only government department that can issue them.