How can I find out if my name is on or not on a birth certificate?

Ask the department of vital statistics. If your name is on the BC you have a right to a copy.

Go to the hospital where ever the baby was born and they should let you know there

The above answer is incorrect... vital statistics maintains birth records... the hospital simply registers the birth. I would look up the laws for your particular location. In many places, you cannot be on the birth certificate unless you were married to the mother or you have signed a paternity acknowledgment. If this is the case in your state, and neither of the above is true, the answer is no.