The New York State DMV site notes that if your registration has past its renewal date then they cannot extend your current registration. They go on to say that you cannot operate a vehicle if the registration is expired. You can renew a registration up to one year after the registration expires though.
You will need to contact the NYS DMV web site for information about how to renew your registration if it has been expired for less than 1 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 NY 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.
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From the information we could find the penalties for a violation of this NY 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. If the violation consists of failure to renew a registration which was valid within 60 days, the fine shall be not less than $40.
[Let CarInsurance.com help you find affordable auto insurance now.] The NYS DMV will be able to give you more information or if you were cited for this offense the clerk of the court should be able to give you more information on penalties and if they would be reduced if you renew your registration before your court date. | 
