In some countries there is a centralized registry of birth certificate records. This is not the case in the USA.
Birth records are maintained individually by each state. Here they are maintained either by the Health Department or the Bureau of Vital Statistics which is usually a department within the Health Department.
Your birth certificate may be stored by your country of birth or even your local town clerk, but not always. These records are often maintained only for genealogical purposes. One of the best ways to track your genealogy, by the way, is a site like ancestry.com. This site maintains birth records taken from state records and from dedicated supporters of genealogy.
Half a century back it was common to see birth records maintained locally. This continued till the need was felt to maintain birth certificate records in a more secure, centralized manner. This need was also the offshoot of population growth and migration. In addition the occurrence of local natural disasters made secure, centralized storage of birth certificate records necessary.
Surprisingly before the 20th century birth certificate records were kept only in the family bible or were not kept at all. Only after the creation of the Social Security Administration in 1935 was the need to keep a birth certificate recognized.
Hospital records often have a copy of your birth certificate. However, this is not a surety. The older you are, the less likely this is. In addition older records do not get transferred to a digital format, so the hospital authorities might not even have the time to search for your record. Here a distinction must be made – at the state level your birth is a matter of official record whereas at the hospital it is merely a matter of medical record.
The census bureau has never maintained a record of people’s birth. However, some people think that it is the job of the census bureau to maintain birth certificate records. As a result the census bureau regularly gets requests for the same. Though older census records show people’s names addresses and age, it is not the job of the census bureau to maintain birth certificate records.
If you want to find the birth certificate records of someone, the place to look is the state of your birth since that is where birth certificate records are officially filed.