Most counties issue a "License to carry is hereby granted" type license. The license itself does not grant your right to carry the firearm concealed, but the laws and regulations of NY do. When you go to fill ou tthe application, look near the top of the form and it should ask you why a license is required. I would write: target shooting, hunting, personal protection, instruction, and anything else you can think of. The license CANNOT be denied (legally) for any reason other than having a negative mental health check, police record, domestic violence issues, or a felony conviction. You should make sure you have enough references that are not going to torpedo your chances either. Make sure they are decent people too, I knew a guy that had five references and two of them had felony DWI's. Not a good thing.
Also: get anything your county has on safe firearms classes or instruction in your area. Take what you can. The more you learn firsthand, the better off you will be. I would start by looking up section 265 and section 400 of the NY CPL and read them. The get into an NRA Basic Pistol class or something equal. Get the information from qualified, knowing sources, not the 'range commandos' you will undoubtedly find.
Gun stores can be a good source for information too. Find a good one in your area and become a good customer. He will be more than willing to helo you out when you do the same for him.
Good luck. Get it started now as it may take a while to get the permit back.