They usually don't just send it to you. At the hospital you fill out the form with the info for the birth certificate so that the hospital can submit it to the county. But if you want it, you have to contact the county and pay them for copy. The hospital should have given you info on how to obtain a copy (usually the dept of vital statistics or something like that). I'd suggest ordering 2 or 3 copies just in case!
I thought it is the township where you delivered. Ex I had my son at Centrastate in freehold... I had to go to freehold township court house/ town hall for them to send a copy.
Re: Birth certificate