There are three types of certificates that acknowledge your citizenship in the United States. More specifically, you may possess one of the following documents: Certificate of Naturalization, Certificate of Citizenship, or Certificate of Repatriation.
http://us-immigration-info.com/who-is-eligible-to-...ation-or-citizenship-document/
When can you apply for a replacement of your certificate? There are basically two grounds for requesting a new certificate of citizenship, naturalization, or repatriation:

Your certificate was lost, damaged, or stolen.
Your name has changed or any other data was incorrectly stated on your certificate.
In both cases you need to make sure that you have enough evidence to prove that your request for a new certificate is valid.