You can go to the Bureau of Vital Statistics at 124 Grand Street (in the Multi-Service Center) and get it during business hours or mail in for it with the downloadable form.
Dogs who have attained 7 months of age or have their adult teeth must be licensed every year. Dog owners must provide proof that their dog has been vaccinated by a licensed veterinarian against rabies. The vaccination must be good for at least 10 of the 12 months in the licensing cycle (January – December).
License fees are $10.00 for spayed/neutered dogs and $14.00 for non-spayed/non-neutered dogs.
See the application here