Birth, death, and marriage records are collected and maintained at the local town/city level by the clerks of each respective location where the events occurred. Check the County Directory (PDF) for the appropriate contact.
For example, if you live in Campbell, but were born at Corning Hospital's old location, you can obtain a copy of your birth certificate through the City of Corning Clerk. If you live in Bath, but were born in the new Corning Hospital location, you can get a copy from the Town of Corning Clerk.
Birth certificates can also be obtained through the NYS Department of Health (DOH). Visit the NYSDOH website for more information. There is a cost associated with getting a copy at either the local clerk's office or the DOH, but it has been found to cost less through the local clerk.