You can register to vote or find out more information about the voting process by either visiting the New York City Board of Elections or the New York State Board of Elections websites. To register to vote in the City of New York, you must be: A citizen of the United States (includes those persons born in Puerto Rico, Guam, and the U.S. Virgin Islands), a New York City resident for at least 30 days, 18 years of age before the next election, not currently serving a jail sentence or be on parole for a felony conviction (unless you have received a pardon for voting purposes), not be adjudged mentally incompetent by a court, and not claim the right to vote elsewhere (oustide the City of New York).
You can also download a NYS Voter Registration Form.