Usually in the description of every apartment there is information regarding the pet policy for that specific apartment. Most owners / managing companies don't like pets on their apartments due to the fact that it can cost a lot to fix any damages caused to the floors. Others are more flexible and are OK with cats while dogs are in a case by case situation. Considering that a pet can make or break a deal, this information is essencial and should be disclosed from the beginning.
The managing companies that accept pets usually have some restrictions and additional fees such as additional rent and security. Some companies can ask you if the pet has been registered in NYC and will request a photo for review and consideration.
When there is not information regarding pets in the ads in some cases this could mean that the policy is flexible but the best advice is to ask that question to your real estate agent before you go ahead with an application, after all the pets need to be legal residents of the apartments as well.
If you are looking for affordable apartments with pet flexible policies anywhere in NYC, I will be glad to help you with your search.
Licensed Real Estate Agent
Ardor New York Real Estate
My current listings: http://www.ardorny.com/nelson-leon