Hi Yuriken56, you have a good heart for taking care of the stray cat. Probably the best thing would be for someone to adopt the stray cat and make it an indoor-only kitty, but in the short term you can provide water, food, and some sort of shelter outside your house.Water
If you could set out fresh water daily, it would be great for the cat, as reliable fresh water may be hard to find outside.Food
Cats have evolved as meat-eaters, so although many cats like starchy foods, vegetables, and even ice cream, most of what they eat should be derived from meat or meat products. This means that the healthiest thing is cat food - Iams is good and widely available. The stray's hunting ability may or may not be good enough for survival. See "Choosing the Right Food for Your Cat" http://www.thecatsite.com/cat_health...p?a=principlesShelter
Since cats are by nature creatures of the night, napping in your house during the day and going out at night should suit the cat's internal clock OK. However, you can still set up a shelter outside for the cat. Here is a way to make a little winter cat bungalow with just a styrofoam box and some household items: http://www.all-creatures.org/ak/feral-shelter.html
Styrofoam is good because it's waterproof and retains heat well.
Also, you can give the cat a quick check to see how healthy he/she is: http://www.thecatsite.com/cat_health....php?a=healthy
Once the immediate survival basics of water, food, and shelter are taken care of, you should try and determine if someone has lost the cat and is looking for him/her. You didn't mention finding a collar on the cat, but many cats have an ID chip implanted between their shoulders - a vet or animal shelter would use a chip scanner to look up the registered owner. You may have an animal-care organization near you that can help. Here is a list of animal shelters in the USA & Canada: http://www.saveourstrays.com/no-kill.htm
If the cat is indeed abandoned, you can move on to the long-term issues of health care, spay/neuter, and adoption/feral management.Health Care
The cat should get a basic veterinary check-up for both common (fleas, tapeworms, etc.) and chronic (Feline Leukemia Virus, etc.) ailments. See if you can find a local veterinarian to work with - here is a thread about vets as a good resource: http://www.thecatsite.com/forums/sho...threadid=22260Spay/Neuter
Due to the cat population crisis, you should ensure that the cat is spayed/neutered. Many vets and shelters will spay/neuter strays and ferals at low cost or for free. Here is a thread with good spay/neuter resources: http://www.thecatsite.com/forums/sho...threadid=16045
And here is an article discussing spay/neuter:http://www.thecatsite.com/cat_care/care.php?a=spayAdoption
It sounds like you and the stray cat are getting fond of each other, but you would have to convince your family to live with him/her as well. And you would need to find out some more about the ups & downs of having a cat, so that you can make an informed decision about whether or not to adopt. It's a potential 15-20 year commitment to providing food, health care, play toys, head rubs, etc. - somewhat comparable to raising a child. That said, if you can work it out, this cat could become one of the joys of your life. Read some of the threads in the "Cats S.O.S." and "Behavior" forums for a perspective on other people's experiences with taking in stray cats.
If you can find a good animal shelter or rescue network near you, they can provide you with a wide range of help; anything from feeding advice to adoption placement to discounted vet care. If you don't find any through the animal shelters link above, you can try asking people on the "Cats S.O.S." forum if they know of any in your area.Feral Cat Management
This can be done quite well, should you determine that this would be your best course of action. You can find out more by reading in the "Feral Colonies" forum.
I hope things work out the best for you and the cat. There's a lot of knowledge, experience, and friendly support here on these forums, so please feel free to post any specific questions and update us on what happens.