The New York State Department of Motor Vehicles site notes that if your registration has passed its renewal date then it cannot extend your current registration. The DMV goes on to say that you cannot operate a vehicle if its registration is expired. You can renew a registration up to one year after the registration expires though.
You need to contact the New York DMV for information on how to renew your registration if it has been expired for less than one year. As for driving without current registration, this is an offense which you can be cited for by law enforcement.
Driving without registration, or with expired registration, in New York State is a violation of New York Vehicle and Traffic Code section 401. It is a separate violation of law to display on your vehicle an expired registration sticker. If you cannot provide valid vehicle registration, the police can issue a ticket for an unregistered vehicle.
From the information we could find, a violation of this New York law can be punishable by a fine of not less than $75 nor more than $300 or by imprisonment for not more than 15 days, or both a fine and jail time. If the violation consists of failure to renew a registration which was valid within 60 days, the fine shall be not less than $40.
If you have vehicle registration that has expired over 60 days ago, but haven't been cited for driving an unregistered vehicle, the penalty appears to be a fine of between $50 and $200.
If your registration has expired, contact your local DMV immediately to correct this issue, since police can pull you over if they find you driving without valid registration. If you have already been cited by law enforcement for this offense, then contact the court listed on your ticket for more information on penalties you could receive and if they would be reduced if you renew your registration before your court date.
Your driving record is a rating factor used by all car insurance companies. Some auto insurance providers won't rate on minor violations like an expired registration ticket, but some will -- especially if your state finds it to be a moving violation. Still it's best to keep a clean driving record without any types of violations on it if possible. (See "Keep your license clean -- and your car insurance cheap")
If this (or any other traffic ticket) affects your car insurance rates with your current car insurance carrier, then base rates, so by shopping around you could save hundreds, if not thousands, off your annual auto insurance premium. (See "3 ways to save big on car insurance"). Insurance companies rate tickets different and have different