No father is named on my birth certificate, when I asked my parents about this they said, we are both your parents.
If I was adopted how can I prove this, would it have been registered.
If I was not adopted does this mean that I should be using my mothers name?

Your legal name is the one on the birth certificate ( in US) and you can not use any other name for official records such as school and legal matters.

If you were adopted, there would be adoption paper work and in the US a new birth certificate issues.