Most towns have an animal shelter where stray dogs and cats can be brought, and from which they can be adopted by individuals and families.
While it was once the case that most animal shelters were run by local governments, and that the fate of animals that ended up in the "pound" was uncertain and not necessarily happy (think "Lady and the Tramp"), there are now many privately-run organizations that are "no-kill" shelters and that take a lot of time to make sure that they match the right animal to the right family, and to make sure that the animals are neutered or spayed in order to keep the stray population under control.
There are also breed-specific rescue leagues throughout the country. If you are looking to adopt a pet from a shelter or rescue league, be patient: you'll have a lot of forms to fill out and questions to answer, but it is all for the good of the animal, and to help you make sure that you are making a commitment that you can sustain.
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