FINDING A CAT
If you find a friendly stray cat and you can't keep it, your first reaction is probably to call a local shelter or rescue group. Don't be surprised when the shelter says it's full, or puts you on a waiting list. There are simply too many stray animals and not enough homes.

The city municipal shelter (in NYC this is the ACC, Animal Care & Control) most likely will accept the cat, but will put it to sleep if it's not adopted within a few days. So, it's understandable that you'd prefer to call a no-kill rescue group. However, ALL shelters and rescue groups have severely limited space, resources, money, and volunteers. So before picking up the phone, try to adopt the cat out yourself. A friendly cat has a good chance of finding a home if you're willing to put a little work into it.

WHAT TO DO RIGHT AWAY

Bring the cat inside.
Keep the stray separated from your other pets (the bathroom will do) until it's tested for FIV and leukemia.
Make an appointment with a vet. The cat will need FIV and leukemia testing, as well as vaccinations and spaying/neutering. Many vets will give you a discount if you explain the situation and discuss your financial limitations. You can also check our list of low-cost veterinary services in the NYC area.

SPREAD THE WORD

Take photos of the cat. Print a poster letting people know the cat is available for adoption. Include your phone number or email address in tear-off strips at the bottom of the poster, and hang them around your neighborhood, in local businesses, at pet stores, and at vet offices. The more posters, the better your chances of finding a home for the stray.
Use social media. Post the cat's photo and information on all your social networks and ask friends to share widely.
Email your contacts. Send photos and information to everyone on your contact list, and again, ask that they forward to their contacts. If you're a member of a neighborhood listserv or Facebook group, be sure to post your message there, too.

INTERVIEW POTENTIAL ADOPTERS

If you've done all of the above, chances are you will get at least one phone call or message from someone interested in adopting the cat. Speak to them on the phone to be sure their interest is serious and they are prepared to provide a forever home for the cat. No sense rescuing a stray, only to have it put back on the street. Ask the following questions:
Why do you want a cat right now?
Have you owned a cat or other pets before? Do you have any pets currently?
Are you prepared to provide proper medical care for the cat, as well as food, litter, toys, and companionship?
Does your building allow cats?
Do you have screens on your windows?
A special note about kittens: kittens have a much easier time finding a home than older cats. Be especially thorough when interviewing people who express interest in kittens. They grow up quickly, so you need to be confident the adopter wants a cat, not just a kitten. And remember, kittens (and therefore owners) will be happier if placed in pairs.

ARRANGE A VISIT

If you're comfortable with the potential adopter after your phone interview, invite him or her to your home to meet the cat. The cat will be more relaxed with a stranger if the meeting takes place in a familiar setting.
If it seems like a good match, you can offer to deliver the cat to the adopter's home. This will be more convenient for them, and gives you a chance to check the home environment (be sure to verify that windows are screened).

LAST RESORT

If, after spreading the word and interviewing potential adopters, you still haven't found a home for the cat, a no-kill shelter will be your last resort.
The fact that you have already had the cat tested, vaccinated, and fixed will vastly increase its chances of being taken in at a no-kill shelter. Offering a donation will help even more. The shelter must provide food, litter, and possibly medical care for the cat until it's adopted. All this costs money, and shelters are doing it for dozens of cats at once.
The NYC Mayor's Alliance for Animals has a list of shelters organized by borough.

VOLUNTEER AT A SHELTER
Another thing you can do to help stray cats is to volunteer at your local shelter. People are always needed to clean cages, foster cats, or walk dogs. Even spending a few hours once a week will make a big impact. The more help a shelter has, the more likely it is that the shelter will be able to take in a cat when you -- or someone else -- finds one.